A New Approach to Braided T-Categories and Generalized Quantum Yang–Baxter Equations

Abstract

We introduce and study a large class of coalgebras (possibly (non)coassociative) with group-algebraic structures Hopf (non)coassociative group-algebras. Hopf (non)coassociative group-algebras provide a unifying framework for classical Hopf algebras and Hopf group-algebras and Hopf coquasigroups. We introduce and discuss the notion of a quasitriangular Hopf (non)coassociative π-algebra and show some of its prominent properties, e.g., antipode S is bijective. As an application of our theory, we construct a new braided T-category and give a new solution to the generalized quantum Yang–Baxter equation.

1. Introduction

Topological quantum field theories (TQFT’s) realize topological invariants of manifolds using ideas from quantum field theory (QFT), see [1,2]. Turaev introduced in [3] a homotopy quantum field theory (HQFT) as a version of a TQFT for manifolds endowed with maps into a fixed topological space and found an algebraic characterization of 2-dimensional HQFT’s whose target space is the Eilenberg–MacLane space $K(\pi ,1)$ determined by a group $\pi$. Furthermore, he established a 3-dimensional HQFT with target space $K(\pi ,1)$ by introducing the notion of a modular $\pi$-category based on a deep connection between the theory of braided categories and invariants of knots, links and 3-manifolds (see [4]). This connection has been essential in the construction of quantum invariants of knots and 3-manifolds from quantum groups, see [2,5].

Turaev proposed the following open problem in [4]; Can one systematically produce interesting modular $\pi$-categories?

Examples of such modular $\pi$-categories can be constructed from the so-called Hopf π-(co)algebras which can be regarded as a generalization of a Hopf algebra, see [6,7,8]. At present, many research works have been done for Hopf $\pi$-(co)algebras, such as Turaev’s Hopf group-coalgebras (cf. [9]), group coalgebra Galois extensions (cf. [10]), Larson-Sweedler theorem (cf. [11]), twisted Drinfel’d doubles (cf. [12]), double construction and Yetter-Drinfel’d modules (cf. [13,14,15]). We mention that a Hopf $\pi$-coalgebra can be regarded as a $\pi$-cograded multiplier Hopf algebra, see [16].

In 2010, Klim and Majid in [17] introduced the notion of a Hopf (co)quasigroup which is a particular case of the notion of an H-bialgebra introduced in [18]. The further research of this mathematical object can be found in the references about many topics, such as Hopf modules (cf. [19]), actions (cf. [20]), twisted smash products (cf. [21]), Yetter–Drinfel’d modules (cf. [22]), and Hopf quasicomodules (cf. [23]).

To highlight Turaev’s achievements on the modular $\pi$-categories, in this article we prefer using the notion of a braided T-category (over $\pi$) appeared in [13] to using a modular $\pi$-category [6]. We will provide a new approach to a braided T-category (over $\pi$) based on the notion of a quasitriangular Hopf (non)coassociative $\pi$-algebra.

An outline of the paper is as follows.

Section 2 provides some preliminary background needed in the paper, such as group-algebras, group-convolution algebras, Hopf group-algebras and Turaev’s braided categories.

In Section 3, we give a new characterization of Hopf group-algebras based on the idea from [24,25]. We mainly prove that $(H,\Delta )$ is a Hopf $\pi$-algebra if and only if $\Delta$ is a $\pi$-algebra homomorphism and the right and left $\pi$-Galois maps are bijective.

In Section 4, we introduce and study the notion of a Hopf non-coassociative $\pi$-algebra which is a large class of coalgebras (possibly non-coassociative) with group-algebraic structures unifying the notions of a classical Hopf algebra, a Hopf $\pi$-algebra and a Hopf coquasigroup. We study its algebraic properties, such as anti-(co)multiplicativity of the antipode.

In Section 5 we mainly study the notion of a crossed Hopf non-coassociative $\pi$-algebra and give some properties of the crossing map. In addition, in Section 6, we discuss the definition and properties of an almost cocommutative Hopf non-coassociative $\pi$-algebra and obtain its equivalent characterization.

In the final section, we will introduce and discuss the definition of a quasitriangular Hopf non-coassociative $\pi$-algebra H and study some main properties of H. We construct a new braided T-category $Re{p}_{\pi }\left(H\right)$ over H.

Throughout the paper, we let $\pi$ be a fixed group and $\mathbb{k}$ be a field (although much of what we do is valid over any commutative ring). We use the Sweedler’s notation to express the coproduct of a coalgebra C as $\Delta \left(c\right)=\sum c_1\otimes c_2$ (cf. [26]).

We set ${\mathbb{k}}^{\ast }=\mathbb{k}\backslash \left\{0\right\}$. All algebras are supposed to be over $\mathbb{k}$ and unitary, but not necessarily associative. The tensor product $\otimes ={\otimes }_{\mathbb{k}}$ is always assumed to be over $\mathbb{k}$. If U and V are $\mathbb{k}$-spaces, ${\sigma }_{U,V}:U\otimes V⟶V\otimes U$ will denote the flip map defined by ${\sigma }_{U,V}(u\otimes v)=v\otimes u$.

We use $i{d}_U$ for the identity map on U, although sometimes, we also write U for this map. We use $i{d}_U^n$ for the map $\underset{n}{\underbrace{id\otimes \cdots \otimes id}}:U\underset{n-1}{\underbrace{\otimes \cdots \otimes }}U⟶U\underset{n-1}{\underbrace{\otimes \cdots \otimes }}U$. The identity element in a quasigroup is denoted by e.

2. Preliminaries

In this section, we recall some basic notions used later, such as group-algebras, group-convolution algebras, Hopf group-algebras and braided T-categories.

2.1. Group-Algebras

We recall the definition of a $\pi$-algebra, following [4]. A $\pi$-algebra (over $\mathbb{k}$) is a family $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ of $\mathbb{k}$-spaces endowed with a family $m={\{m_{\alpha ,\beta }:A_{\alpha }\otimes A_{\beta }⟶A_{\alpha \beta }\}}_{\alpha ,\beta \in \pi }$ of $\mathbb{k}$-linear maps (the multiplication) and a $\mathbb{k}$-linear map $\eta :\mathbb{k}⟶A_1$ (the unit) such that m is associative in the sense that, for any $\alpha ,\beta ,\gamma \in \pi ,$

$$\begin{array}{c}\hfill m_{\alpha \beta ,\gamma }(m_{\alpha ,\beta }\otimes i{d}_{A_{\gamma }})=m_{\alpha ,\beta \gamma }(i{d}_{A_{\alpha }}\otimes m_{\beta ,\gamma });\end{array}$$

(1)

$$\begin{array}{c}\hfill m_{\alpha ,1}(i{d}_{A_{\alpha }}\otimes \eta )=i{d}_{A_{\alpha }}=m_{1,\alpha }(\eta \otimes i{d}_{A_{\alpha }}).\end{array}$$

(2)

Note that $(A_1,m_{1,1},\eta )$ is an algebra in the usual sense of the word.

For all $\alpha ,\beta \in \pi ,h\in A_{\alpha },k\in A_{\beta }$, we write $hk=m_{\alpha ,\beta }(h\otimes k)$. The associativity axiom gives that

$$\left(hk\right)l=h\left(kl\right),\forall \alpha ,\beta ,\gamma \in \pi ,h\in A_{\alpha },k\in A_{\beta },l\in A_{\gamma }.$$

Set $\eta \left(1_{\mathbb{k}}\right)=1$. The unit axiom gives that $h1=h=1h,\forall \alpha \in \pi ,h\in A_{\alpha }.$

For all $\alpha \in \pi$, the $\mathbb{k}$-space $A_{\alpha }$ is called the $\alpha$-th component of A.

A $\pi$-algebra morphism between two $\pi$-algebras A and $A^{\prime }$ (with multiplications m and $m^{\prime }$, respectively) is a family $f={\{f_{\alpha }:A_{\alpha }⟶A_{\alpha }^{\prime }\}}_{\alpha \in \pi }$ of $\mathbb{k}$-linear maps such that $f_{\alpha \beta }{m}_{\alpha ,\beta }=m_{\alpha ,\beta }^{\prime }(f_{\alpha }\otimes f_{\beta })$ and $f_1\left(1\right)=1^{\prime },$ for all $\alpha ,\beta \in \pi$. The $\pi$-algebra isomorphism $f={\{f_{\alpha }:A_{\alpha }⟶A_{\alpha }^{\prime }\}}_{\alpha \in \pi }$ is a $\pi$-algebra morphism in which each $f_{\alpha }$ is a linear isomorphism.

Set ${\overline{A}}_{\alpha }=A_{{\alpha }^{-1}}$ and ${\overline{m}}_{\alpha ,\beta }=m_{{\alpha }^{-1},{\beta }^{-1}}^{op}=m_{{\beta }^{-1},{\alpha }^{-1}}\circ {\sigma }_{H_{{\alpha }^{-1}},H_{{\beta }^{-1}}}$. Then comes a $\pi$-algebra $\overline{A}={\left\{{\overline{A}}_{\alpha }\right\}}_{\alpha \in \pi }$ with the same unit element 1 as in A and the multiplication given by $\overline{m}={\left\{{\overline{m}}_{\alpha ,\beta }\right\}}_{\alpha ,\beta \in \pi }$.

2.2. Group-Convolution Algebra

Let $A={(\left\{A_{\alpha }\right\},m,\eta )}_{\alpha \in \pi }$ be a $\pi$-algebra and $(C,\Delta ,\epsilon )$ be a (not necessarily coassociative) coalgebra with comultiplication $\Delta$ and counit $\epsilon$. For any $f\in {\mathrm{Hom}}_{\mathbb{k}}(C,A_{\alpha })$ and $g\in {\mathrm{Hom}}_{\mathbb{k}}(C,A_{\beta })$, we define their convolution product by

$$f\ast g=m_{\alpha ,\beta }(f\otimes g)\Delta \in {\mathrm{Hom}}_{\mathbb{k}}(C,A_{\alpha \beta }).$$

(3)

Using Equation (3), one verifies that the $\mathbb{k}$-space

$$\mathrm{Conv}(C,A)={\displaystyle \underset{\alpha \in \pi }{⨁}}{\mathrm{Hom}}_{\mathbb{k}}(C,A_{\alpha })$$

endowed with the convolution product ∗ and the unit element $\epsilon 1$, is called π-convolution algebra, which is not necessarily a coassociative $\pi$-graded algebra.

In particular, for $C=\mathbb{k}$, the associative $\pi$-graded algebra $\mathrm{Conv}(C,A)={\displaystyle \underset{\alpha \in \pi }{⨁}}{\mathrm{Hom}}_{\mathbb{k}}(\mathbb{k},A_{\alpha })={\displaystyle \underset{\alpha \in \pi }{⨁}}{A}_{\alpha }$ is denoted by $A_{\ast }$.

2.3. Hopf Group-Algebras

Recall from [3] that a Hopf group-algebra over $\pi$ is a $\pi$-algebra $H={(\left\{H_{\alpha }\right\},m={\{m_{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha \beta }\}}_{\alpha ,\beta \in \pi },\eta )}_{\alpha \in \pi }$, endowed with a family $S={\{S_{\alpha }:H_{\alpha }⟶H_{{\alpha }^{-1}}\}}_{\alpha \in \pi }$ of $\mathbb{k}$-linear maps (the antipode) such that the following conditions hold:

$$\begin{array}{c}\mathrm{each}(H_{\alpha },{\Delta }_{\alpha },{\epsilon }_{\alpha })\mathrm{is} \mathrm{a} \mathrm{counital} \mathrm{coassociative} \mathrm{coalgebra}\\ \mathrm{with} \mathrm{comultiplication}{\Delta }_{\alpha }\mathrm{and} \mathrm{counit} \mathrm{element}{\epsilon }_{\alpha };\end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathrm{for} \mathrm{all}\alpha ,\beta \in \pi ,& \eta :\mathbb{k}⟶H_{1}and{m}_{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha \beta }\hfill \\ \hfill & \mathrm{are} \mathrm{coalgebra} \mathrm{homomorphisms},\hfill \end{array}$$

(5)

$$\mathrm{for} \mathrm{all}\alpha \in \pi ,m_{{\alpha }^{-1},\alpha }(S_{\alpha }\otimes i{d}_{H_{\alpha }}){\Delta }_{\alpha }=1{\epsilon }_{\alpha }=m_{\alpha ,{\alpha }^{-1}}(i{d}_{H_{\alpha }}\otimes S_{\alpha }){\Delta }_{\alpha }.$$

(6)

Let $H=({\{H_{\alpha },{\Delta }_{\alpha },{\epsilon }_{\alpha }\}}_{\alpha \in \pi },m,\eta ,S)$ be a Hopf $\pi$-algebra. Then

$$\begin{array}{cc}\hfill S_{\alpha \beta }\left(ab\right)& =S_{\beta }\left(b\right)S_{\alpha }\left(a\right),\forall \alpha ,\beta \in \pi ,a\in H_{\alpha },b\in H_{\beta };\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill S_1\left(1\right)& =1;\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\Delta }_{{\alpha }^{-1}}{S}_{\alpha }& ={\sigma }_{H_{{\alpha }^{-1}},H_{{\alpha }^{-1}}}(S_{\alpha }\otimes S_{\alpha }){\Delta }_{\alpha },\forall \alpha \in \pi ;\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\epsilon }_{{\alpha }^{-1}}{S}_{\alpha }& ={\epsilon }_{\alpha },\forall \alpha \in \pi .\hfill \end{array}$$

(10)

2.4. Braided T-Categories

Let $\pi$ be a group. A pre-T-category $\mathcal{T}$ (over π) is given by the following datum:

• A tensor category $\mathcal{T}$.
• A family of sub categories ${\left\{{\mathcal{T}}_{\alpha }\right\}}_{\alpha \in \pi }$ such that $\mathcal{T}$ is a disjoint union of this family and that $U\otimes V\in {\mathcal{T}}_{\alpha \beta }$, for any $\alpha ,\beta \in \pi ,U\in {\mathcal{T}}_{\alpha },$ and $V\in {\mathcal{T}}_{\beta }$.
  Furthermore, $\mathcal{T}=\left\{{\mathcal{T}}_{\alpha }\right\}$ satisfies the following condition:
• Denote by $aut\left(\mathcal{T}\right)$ the group of the invertible strict tensor functors from $\mathcal{T}$ to itself, a group homomorphism $\phi :\pi ⟶aut\left(\mathcal{T}\right):\beta \mapsto {\phi }_{\beta },$ the conjugation such that ${\phi }_{\beta }\left({\mathcal{T}}_{\alpha }\right)={\mathcal{T}}_{\beta \alpha {\beta }^{-1}}$ for any $\alpha ,\beta \in \pi$. Then we call $\mathcal{T}$ a crossed T-category.

We will use the left index notation in Turaev: Given $\beta \in \pi$ and an object $V\in {\mathcal{T}}_{\beta }$, the functor ${\phi }_{\beta }$ will be denoted by ${}^V(·)$ or ${}^{\beta }(·)$. We use the notation ${}^{\overline{V}}(·)$ for ${}^{{\beta }^{-1}}(·)$. Then we have ${}^{V}i{d}_U=i{d}_{V^U}$ and ${}^V(g\circ f)={}^{V}g\circ {}^{V}f$. We remark that since the conjugation $\phi :\pi ⟶aut\left(\mathcal{T}\right)$ is a group homomorphism, for any $V,W\in \mathcal{T}$, we have ${}^{V\otimes W}(·)={}^V{(}^W(·))$ and ${}^1(·)={}^V{(}^{\overline{V}}(·))={}^{\overline{V}}{(}^V(·))=i{d}_{\mathcal{T}}$ and that since, for any $V\in \mathcal{T}$, the functor ${}^V(·)$ is strict, we have ${}^V(f\otimes g)={}^{V}f\otimes {}^{V}g$, for any morphism f and g in $\mathcal{T}$, and ${}^{V}1=1$. In addition, we will use $\mathcal{T}(U,V)$ for a set of morphisms (or arrows) from U to V in $\mathcal{T}$.

Recall from [13] or [6] that a braided T-category (over $\pi$) is a crossed T-category $\mathcal{T}$ endowed with a braiding, i.e., with a family of isomorphisms

$$c={\{c_{U,V}\in \mathcal{T}(U\otimes V,\left({}^{U}V\right)\otimes V)\}}_{U,V\in \mathcal{T}}$$

satisfying the following conditions:

⋅

for any arrow $f\in {\mathcal{T}}_{\alpha }(U,U^{{}^{\prime }})$ with $\alpha \in \pi$, $g\in {\mathcal{T}}_{\beta }(V,V^{{}^{\prime }})$, we have

$$(\left({}^{\alpha }g\right)\otimes f)\circ c_{U,V}=c_{U^{{}^{\prime }},V^{{}^{\prime }}}\circ (f\otimes g);$$

⋅

for all $U,V,W\in \mathcal{T}$, we have

$$c_{U\otimes V,W}=a_{{}^{U\otimes V}W,U,V}\circ (c_{U,{}^{V}W}\otimes i{d}_V)\circ a_{U,{}^{V}W,V}^{-1}\circ (i{d}_U\otimes c_{V,W})\circ a_{U,V,W},$$

(11)

$$c_{U,V\otimes W}=a_{{}^{U}V,{}^{U}W,U}^{-1}\circ (i{d}_{{}^{U}V}\otimes c_{U,W})\circ a_{{}^{U}V,U,W}\circ (c_{U,V}\otimes i{d}_W)\circ a_{U,V,W}^{-1};$$

(12)

⋅

for any $U,V\in \mathcal{T}$, $\alpha \in \pi$, ${\phi }_{\alpha }\left(c_{U,V}\right)=c_{{\phi }_{\alpha }\left(U\right),{\phi }_{\alpha }\left(V\right)}$.

3. A New Characterization of Hopf Group-Algebras

Based on the idea from [24,25], in this section we mainly show that H is a Hopf $\pi$-algebra if and only if $\Delta$ is a $\pi$-algebra homomorphism and the right and left $\pi$-Galois maps both have inverses.

Proposition 1.

If H is a Hopf π-algebra, then the families of linear maps $T_1=\{T_1^{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha }\otimes H_{\alpha \beta }\}$ (called the left π-Galois map) and $T_2=\{T_2^{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha \beta }\otimes H_{\beta }\}$ (called the right π-Galois map), defined, respectively, by

$$T_1^{\alpha ,\beta }(a\otimes b)={\Delta }_{\alpha }\left(a\right)(1\otimes b) and T_2^{\alpha ,\beta }(a\otimes b)=(a\otimes 1){\Delta }_{\beta }\left(b\right)$$

are bijective.

Proof. 

Define two families of linear maps

$$R_1=\{R_1^{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha }\otimes H_{{\alpha }^{-1}\beta }\},\mathrm{and}{R}_2=\{R_2^{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha {\beta }^{-1}}\otimes H_{\beta }\},$$

respectively, by

$$R_1^{\alpha ,\beta }(a\otimes b)=\left((i{d}_{H_{\alpha }}\otimes S_{\alpha }){\Delta }_{\alpha }\left(a\right)\right)(1\otimes b)\mathrm{and}{R}_2^{\alpha ,\beta }(a\otimes b)=(a\otimes 1)\left((S_{\beta }\otimes i{d}_{H_{\beta }}){\Delta }_{\beta }\left(b\right)\right).$$

By a straightforward application of the properties of S one can show that $R_1^{\alpha ,\alpha \beta }$ is the inverse of $T_1^{\alpha ,\beta }$ and that $R_2^{\alpha \beta ,\beta }$ is the inverse of $T_2^{\alpha ,\beta }$. □

If the antipode S has an inverse, then also the other families of linear maps, defined by

$$T_3^{\alpha ,\beta }(a\otimes b)={\Delta }_{\alpha }\left(a\right)(b\otimes 1) \mathrm{and} T_4^{\alpha ,\beta }(a\otimes b)=(1\otimes a){\Delta }_{\beta }\left(b\right)$$

are bijections. This follows, e.g., from the fact that $S^{cop}={\{S_{\alpha }^{cop}=S_{{\alpha }^{-1}}^{-1}\}}_{\alpha \in \pi }$ will be the antipode if we set $H_{\alpha }^{cop}=H_{\alpha }$ as an algebra and replace ${\Delta }_{\alpha }$ by the opposite comultiplication ${\Delta }_{\alpha }^{cop}={\sigma }_{H_{\alpha },H_{\alpha }}{\Delta }_{\alpha }$.

We now discuss some results if $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$ is a unital, but not necessarily associative, group algebra over $\mathbb{k}$ with a family $\Delta ={\{{\Delta }_{\alpha }:H_{\alpha }⟶H_{\alpha }\otimes H_{\alpha }\}}_{\alpha \in \pi }$ of coassociative comultiplications, a family of linear maps, such that the families of linear maps $T_1$ and $T_2$ are bijections.

Define a family of maps $E={\{E_{\alpha }:H_{\alpha }⟶H_1\}}_{\alpha \in \pi }$ by

$$E_{\alpha }\left(a\right)b=m_{\alpha ,{\alpha }^{-1}\beta }{\left(T_1^{\alpha ,{\alpha }^{-1}\beta }\right)}^{-1}(a\otimes b)$$

where $m_{\alpha ,{\alpha }^{-1}\beta }$ denotes multiplication, considered as a linear map from $H_{\alpha }\otimes H_{\beta }$ to $H_{\beta }$ and where $T_1^{\alpha ,{\alpha }^{-1}\beta }$ is defined as before by $T_1^{\alpha ,{\alpha }^{-1}\beta }(a\otimes b)={\Delta }_{\alpha }\left(a\right)(1\otimes b)\in H_{\alpha }\otimes H_{\beta }$.

Lemma 1.

For all $a,b\in H_{\alpha }$, we have $(H_{\alpha \beta }\otimes E_{\beta })\left((a\otimes 1){\Delta }_{\beta }\left(b\right)\right)=ab\otimes 1$.

Proof. 

By the coassociativity of ${\Delta }_{\alpha }$, one can easily obtain

$$\begin{array}{cc}& (m_{\alpha ,\beta }\otimes H_{\beta }\otimes H_{\beta \gamma })(H_{\alpha }\otimes {\Delta }_{\beta }\otimes H_{\beta \gamma })(H_{\alpha }\otimes H_{\beta }\otimes m_{\beta ,\gamma })(H_{\alpha }\otimes {\Delta }_{\beta }\otimes H_{\gamma })\hfill \\ \hfill =& (H_{\alpha \beta }\otimes H_{\beta }\otimes m_{\beta ,\gamma })(H_{\alpha \beta }\otimes {\Delta }_{\beta }\otimes H_{\gamma })(m_{\alpha ,\beta }\otimes H_{\beta }\otimes H_{\gamma })(H_{\alpha }\otimes {\Delta }_{\beta }\otimes H_{\gamma }).\hfill \end{array}$$

Assume $a\in H_{\alpha },b\in H_{\beta }$ and, since $T_1$ is surjective, let

$$a\otimes b=\sum _{i=1}^n{\Delta }_{\alpha }\left(a_i\right)(1\otimes b_i).$$

If we apply ${\Delta }_{\alpha }\otimes H_{\beta }$ and then multiply with $c\otimes 1\otimes 1$ to the both sides of the above equation, where $c\in H_{\gamma }$, on the left, by the direct conclusion of coassociativity given above, we can obtain

$$(c\otimes 1){\Delta }_{\alpha }\left(a\right)\otimes b=T_1^{\alpha ,{\alpha }^{-1}\beta }\left(\sum (\phi \otimes H_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a_i\right)\right)\otimes b_i\right).$$

By the definition of $E_{\alpha }$ we get

$$E_{\alpha }\left((\phi \otimes H_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a\right)\right)\right)b=\sum (\phi \otimes H_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a_i\right)\right)b_i.$$

So

$$\begin{array}{cc}\hfill \phantom{=}(\phi \otimes H_{\beta })& \left((H_{\gamma \alpha }\otimes E_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a\right)\right)(1\otimes b)\right)=E_{\alpha }\left((\phi \otimes H_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a\right)\right)\right)b\hfill \\ & =\sum (\phi \otimes H_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a_i\right)\right)b_i\hfill \\ & =(\phi \otimes H_{\beta })\left((c\otimes 1)\sum {\Delta }_{\alpha }\left(a_i\right)(1\otimes b_i)\right)\hfill \\ & =(\phi \otimes H_{\beta })\left((c\otimes 1)(a\otimes b)\right).\hfill \end{array}$$

Because this holds for all $\phi$, we get

$$(H_{\gamma \alpha }\otimes E_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a\right)\right)(1\otimes b)=(ca\otimes 1)(1\otimes b).$$

This gives the required formula. □

Lemma 2.

$E_{\alpha }\left(H_{\alpha }\right)\subseteq \mathbb{k}1$.

Proof. 

By the surjectivity of $T_2$, defined by $T_2^{\alpha {\beta }^{-1},\beta }(x\otimes y)=(x\otimes 1){\Delta }_{\beta }\left(y\right)\in H_{\alpha }\otimes H_{\beta }$, we see that $a\otimes E_{\beta }\left(b\right)\in H_{\alpha }\otimes 1$ for all a in $H_{\alpha }$ and b in $H_{\beta }$. This gives the result. □

Define a family of linear maps

$$\epsilon ={\{{\epsilon }_{\alpha }:H_{\alpha }⟶\mathbb{k}\}}_{\alpha \in \pi }$$

by ${\epsilon }_{\alpha }\left(a\right)1=E_{\alpha }\left(a\right)$.

Remark 1.

The formula in Lemma 1 can be rewritten as

$$(H_{\alpha \beta }\otimes {\epsilon }_{\beta })\left((a\otimes 1){\Delta }_{\beta }\left(b\right)\right)=ab.$$

It can be concluded that the associativity of $m={\{m_{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha \beta }\}}_{\alpha ,\beta \in \pi }$ holds if and only if $(H_{\alpha ,\beta \gamma }\otimes {\epsilon }_{\beta \gamma })\left((a\otimes 1){\Delta }_{\beta \gamma }\left(bc\right)\right)=(H_{\alpha \beta }\otimes {\epsilon }_{\beta })\left((a\otimes 1){\Delta }_{\beta }\left(b\right)\right)c$.

By the definition of $\epsilon$, we also get

$$({\epsilon }_{\alpha }\otimes H_{\beta })(a\otimes b)=E_{\alpha }\left(a\right)b=m_{\alpha ,{\alpha }^{-1}\beta }{\left(T_1^{\alpha ,{\alpha }^{-1}\beta }\right)}^{-1}(a\otimes b)$$

and, by the surjectivity of $T_1$; hence,

$$({\epsilon }_{\alpha }\otimes H_{\alpha \beta })\left({\Delta }_{\alpha }\left(x\right)(1\otimes y)\right)=xy.$$

These formulas just mean

$$(H_{\alpha }\otimes {\epsilon }_{\alpha }){\Delta }_{\alpha }=H_{\alpha }=({\epsilon }_{\alpha }\otimes H_{\alpha }){\Delta }_{\alpha }.$$

It shows that, for any $\alpha \in \pi$, $(H_{\alpha },{\Delta }_{\alpha },{\epsilon }_{\alpha })$ is a coalgebra.

Define a family of maps $F={\{F_{\alpha }:H_{\alpha }⟶H_1\}}_{\alpha \in \pi }$ by $a{F}_{\beta }\left(b\right)=m_{\alpha {\beta }^{-1},\beta }{\left(T_2^{\alpha {\beta }^{-1},\beta }\right)}^{-1}(a\otimes b)$ where $m_{\alpha {\beta }^{-1},\beta }$ denotes multiplication, considered as a linear map from $H_{\alpha {\beta }^{-1}}\otimes H_{\beta }$ to $H_{\alpha }$ and where $T_2^{\alpha {\beta }^{-1},\beta }$ is defined as before by $T_2^{\alpha {\beta }^{-1},\beta }(a\otimes b)=(a\otimes 1){\Delta }_{\beta }\left(b\right)$.

Similar to Lemmas 1 and 2, we have

Lemma 3.

For all $a\in H_{\alpha }$ and $b\in H_{\beta }$, we have $(F_{\alpha }\otimes H_{\alpha \beta })\left({\Delta }_{\alpha }\left(a\right)(1\otimes b)\right)=1\otimes ab$.

Lemma 4.

$F_{\alpha }\left(H_{\alpha }\right)\subseteq \mathbb{k}1$.

Define a family of linear maps

$$ϵ={\{{ϵ}_{\alpha }:H_{\alpha }⟶\mathbb{k}\}}_{\alpha \in \pi }$$

by ${ϵ}_{\alpha }\left(a\right)1=F_{\alpha }\left(a\right)$.

Remark 2.

The formula in Lemma 3 can be rewritten as

$$({ϵ}_{\alpha }\otimes H_{\alpha \beta })\left({\Delta }_{\alpha }\left(a\right)(1\otimes b)\right)=ab.$$

It can be concluded that the associativity of $m={\{m_{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha \beta }\}}_{\alpha ,\beta \in \pi }$ holds if and only if $({ϵ}_{\alpha \beta }\otimes H_{\alpha \beta \gamma })\left({\Delta }_{\alpha \beta }\left(ab\right)(1\otimes c)\right)=a({ϵ}_{\beta }\otimes H_{\beta \gamma })\left({\Delta }_{\beta }\left(b\right)(1\otimes c)\right)$.

By the definition of $ϵ$, we also get

$$(H_{\alpha }\otimes {ϵ}_{\beta })(a\otimes b)=a{F}_{\beta }\left(b\right)=m_{\alpha {\beta }^{-1},\beta }{\left(T_2^{\alpha {\beta }^{-1},\beta }\right)}^{-1}(a\otimes b)$$

and, by the surjectivity of $T_2$; hence,

$$(H_{\alpha \beta }\otimes {ϵ}_{\beta })\left((x\otimes 1){\Delta }_{\beta }\left(y\right)\right)=xy.$$

These formulas just mean

$$(H_{\alpha }\otimes {ϵ}_{\alpha }){\Delta }_{\alpha }=H_{\alpha }=({ϵ}_{\beta }\otimes H_{\alpha }){\Delta }_{\alpha }.$$

It shows that, for any $\alpha \in \pi$, $(H_{\alpha },{\Delta }_{\alpha },{ϵ}_{\alpha })$ is a coalgebra.

Due to the loss of the associativity of $m={\left\{m_{\alpha ,\beta }\right\}}_{\alpha ,\beta \in \pi }$, the counit family $\epsilon ={\left\{{\epsilon }_{\alpha }\right\}}_{\alpha \in \pi }$ is not necessarily a $\pi$-algebra homomorphism even when $\Delta ={\left\{{\Delta }_{\alpha }\right\}}_{\alpha \in \pi }$ is a $\pi$-algebra homomorphism. From now on, every group algebra will tacitly be assumed to carry the associativity of its multiplication and we also suppose that $\Delta ={\left\{{\Delta }_{\alpha }\right\}}_{\alpha \in \pi }$ is a $\pi$-algebra homomorphism.

We will show that $\epsilon ={\left\{{\epsilon }_{\alpha }\right\}}_{\alpha \in \pi }$ satisfies the usual properties of the counit family in Hopf group-algebra theory.

Lemma 5.

$\epsilon ={\left\{{\epsilon }_{\alpha }\right\}}_{\alpha \in \pi }$ is a π-algebra homomorphism.

Proof. 

By Lemma 1, we have

$$(H_{\alpha \beta \gamma }\otimes {\epsilon }_{\beta \gamma })\left((a\otimes 1){\Delta }_{\beta \gamma }\left(bc\right)\right)=a\left(bc\right)$$

for all $a\in H_{\alpha },b\in H_{\beta },c\in H_{\gamma }$. Then

$$(H_{\alpha \beta \gamma }\otimes {\epsilon }_{\beta \gamma })\left((a\otimes 1){\Delta }_{\beta }\left(b\right){\Delta }_{\gamma }\left(c\right)\right)=a\left(bc\right)=\left(ab\right)c=(H_{\alpha \beta }\otimes {\epsilon }_{\beta })\left((a\otimes 1){\Delta }_{\beta }\left(b\right)\right)c.$$

By the surjectivity of $T_2$ we get

$$\begin{array}{cc}\hfill (H_{\alpha \gamma }\otimes {\epsilon }_{\beta \gamma })\left((a\otimes b){\Delta }_{\gamma }\left(c\right)\right)& =(H_{\alpha }\otimes {\epsilon }_{\beta })(a\otimes b)c=a{\epsilon }_{\beta }\left(b\right)c={\epsilon }_{\beta }\left(b\right)ac\hfill \\ & ={\epsilon }_{\beta }\left(b\right)(H_{\alpha \gamma }\otimes {\epsilon }_{\gamma })\left((a\otimes 1){\Delta }_{\gamma }\left(c\right)\right)\hfill \end{array}$$

for all $a\in H_{\alpha },b\in H_{\beta },c\in H_{\gamma }$. Again by the surjectivity of $T_2$ we get

$$(H_{\alpha }\otimes {\epsilon }_{\beta \gamma })(a\otimes bc)={\epsilon }_{\beta }\left(b\right)(H_{\alpha }\otimes {\epsilon }_{\gamma })(a\otimes c).$$

This means

$$a{\epsilon }_{\beta \gamma }\left(bc\right)=a{\epsilon }_{\beta }\left(b\right){\epsilon }_{\gamma }\left(c\right).$$

Set $\alpha =1$ and $a=1$, we have

$${\epsilon }_{\beta \gamma }\left(bc\right)={\epsilon }_{\beta }\left(b\right){\epsilon }_{\gamma }\left(c\right).$$

Since $(H_{\alpha }\otimes {\epsilon }_{\alpha }){\Delta }_{\alpha }=H_{\alpha }=({\epsilon }_{\alpha }\otimes H_{\alpha }){\Delta }_{\alpha }$, we obtain

$$1=H_1\left(1\right)=({\epsilon }_1\otimes H_1){\Delta }_1\left(1\right)=({\epsilon }_1\otimes H_1)(1\otimes 1)={\epsilon }_1\left(1\right)1$$

whereby ${\epsilon }_1\left(1\right)=1_{\mathbb{k}}$. □

Remark that, by a similar reasoning, we can also claim that $ϵ={\left\{{ϵ}_{\alpha }\right\}}_{\alpha \in \pi }$ is a $\pi$-algebra homomorphism.

In fact, for all $\alpha \in \pi$, ${\epsilon }_{\alpha }={ϵ}_{\alpha }$. In order to check this result, we need the following lemma.

Lemma 6.

For all $a\in H_{\alpha },b\in H_{\beta }$, $m_{\alpha {\beta }^{-1},\beta }\left({\left(T_2^{\alpha {\beta }^{-1},\beta }\right)}^{-1}(a\otimes b)\right)=a{\epsilon }_{\beta }\left(b\right)$.

Proof. 

Assume $a\in H_{\alpha },b\in H_{\beta }$ and, since $T_2$ is surjective, let

$$a\otimes b=T_2^{\alpha {\beta }^{-1},\beta }\left(\sum _{i=1}^n{a}_i\otimes b_i\right)=\sum _{i=1}^n(a_i\otimes 1){\Delta }_{\beta }\left(b_i\right).$$

Hence,

$$\begin{array}{cc}\hfill & \phantom{=}{m}_{\alpha {\beta }^{-1},\beta }\left({\left(T_2^{\alpha {\beta }^{-1},\beta }\right)}^{-1}(a\otimes b)\right)=m_{\alpha {\beta }^{-1},\beta }\left({\left(T_2^{\alpha {\beta }^{-1},\beta }\right)}^{-1}{T}_2^{\alpha {\beta }^{-1},\beta }\left(\sum _{i=1}^n{a}_i\otimes b_i\right)\right)\hfill \\ \hfill & =m_{\alpha {\beta }^{-1},\beta }\left(\sum _{i=1}^n{a}_i\otimes b_i\right)=\sum _{i=1}^n{a}_i{b}_i=\sum _{i=1}^n{a}_i{b}_i{}_{\left(1\right)}{\epsilon }_{\beta }\left(b_i{}_{\left(2\right)}\right)=(H_{\alpha }\otimes {\epsilon }_{\beta })\sum _{i=1}^n(a_i{b}_i{}_{\left(1\right)}\otimes b_i{}_{\left(2\right)})\hfill \\ \hfill & =(H_{\alpha }\otimes {\epsilon }_{\beta })\sum _{i=1}^n(m_{\alpha {\beta }^{-1},\beta }\otimes H_{\beta })(H_{\alpha {\beta }^{-1}}\otimes {\Delta }_{\beta })(a_i\otimes b_i)=(H_{\alpha }\otimes {\epsilon }_{\beta })(a\otimes b)=a{\epsilon }_{\beta }\left(b\right).\hfill \end{array}$$

□

Lemma 7.

For all $\alpha \in \pi$, ${\epsilon }_{\alpha }={ϵ}_{\alpha }$.

Proof. 

For all $a\in H_{\alpha },b\in H_{\beta }$, by the definition of F, we have

$$a{ϵ}_{\beta }\left(b\right)=a{F}_{\beta }\left(b\right)=m_{\alpha {\beta }^{-1},\beta }\left({\left(T_2^{\alpha {\beta }^{-1},\beta }\right)}^{-1}(a\otimes b)\right).$$

By Lemma 6, we also get

$$m_{\alpha {\beta }^{-1},\beta }\left({\left(T_2^{\alpha {\beta }^{-1},\beta }\right)}^{-1}(a\otimes b)\right)=a{\epsilon }_{\beta }\left(b\right).$$

It follows that $a{ϵ}_{\beta }\left(b\right)=a{\epsilon }_{\beta }\left(b\right)$. □

We have constructed a counit family $\epsilon ={\left\{{\epsilon }_{\alpha }\right\}}_{\alpha \in \pi }$ satisfying the usual properties of the counit family in Hopf group-algebra theory.

We will construct an antihomomorphism $S={\left\{S_{\alpha }\right\}}_{\alpha \in \pi }$ that has the properties of the antipode in the Hopf group-algebra theory.

Definition 1.

Define a family of linear maps $S={\{S_{\alpha }:H_{\alpha }\to H_{{\alpha }^{-1}}\}}_{\alpha \in \pi }$ by

$$S_{\alpha }\left(a\right)b=({\epsilon }_{\alpha }\otimes H_{{\alpha }^{-1}\beta }){\left(T_1^{\alpha ,{\alpha }^{-1}\beta }\right)}^{-1}(a\otimes b)$$

for all $a\in H_{\alpha },b\in H_{\beta }$.

Lemma 8.

$(H_{\gamma \alpha }\otimes S_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a\right)\right)(1\otimes b)=(c\otimes 1){\left(T_1^{\alpha ,{\alpha }^{-1}\beta }\right)}^{-1}(a\otimes b).$

Proof. 

As in the proof of Lemma 1, for $\phi \in H_{\gamma \alpha }^{\ast }$ and $a\in H_{\alpha },b\in H_{\beta },c\in H_{\gamma }$, we get

$$(\phi \otimes H_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a\right)\right)\otimes b=T_1^{\alpha ,{\alpha }^{-1}\beta }(\sum (\phi \otimes H_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a_i\right)\right)\otimes b_i$$

if $a\otimes b={\displaystyle \sum _{i=1}^n}{\Delta }_{\alpha }\left(a_i\right)(1\otimes b_i)$. Then, by the definition of S, we get

$$\begin{array}{cc}\hfill S_{\alpha }\left((\phi \otimes H_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a\right)\right)\right)b& =({\epsilon }_{\alpha }\otimes H_{{\alpha }^{-1}\beta })\left(\sum (\phi \otimes H_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a_i\right)\right)\otimes b_i\right)\hfill \\ & =(\phi \otimes H_{{\alpha }^{-1}\beta })\left(\sum (H_{\gamma \alpha }\otimes {\epsilon }_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a_i\right)\right)\otimes b_i\right)\hfill \\ & =(\phi \otimes H_{{\alpha }^{-1}\beta })\left(\sum c{a}_i\otimes b_i\right)\hfill \\ & =(\phi \otimes H_{{\alpha }^{-1}\beta })\left((c\otimes 1){\left(T_1^{\alpha ,{\alpha }^{-1}\beta }\right)}^{-1}(a\otimes b)\right).\hfill \end{array}$$

Hence,

$$(\phi \otimes H_{{\alpha }^{-1}\beta })\left((H_{\gamma \alpha }\otimes S_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a\right)\right)(1\otimes b)\right)=(\phi \otimes H_{{\alpha }^{-1}\beta })\left((c\otimes 1){\left(T_1^{\alpha ,{\alpha }^{-1}\beta }\right)}^{-1}(a\otimes b)\right).$$

This is true for all $\phi \in H_{\gamma \alpha }^{\ast }$ and hence proves the result. □

Lemma 9.

For all $a\in H_{\alpha },b\in H_{\beta }$ and $c\in H_{\gamma }$, we have

$$m_{\gamma \alpha ,{\alpha }^{-1}\beta }\left((H_{\gamma \alpha }\otimes S_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a\right)\right)(1\otimes b)\right)=c{\epsilon }_{\alpha }\left(a\right)b.$$

Proof. 

We get this formula if we apply $m_{\gamma \alpha ,{\alpha }^{-1}\beta }$ on the equation in Lemma 8 because

$$\begin{array}{cc}\hfill \phantom{=}{m}_{\gamma \alpha ,{\alpha }^{-1}\beta }& \left((H_{\gamma \alpha }\otimes S_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a\right)\right)(1\otimes b)\right)=m_{\gamma \alpha ,{\alpha }^{-1}\beta }\left((c\otimes 1){\left(T_1^{\alpha ,{\alpha }^{-1}\beta }\right)}^{-1}(a\otimes b)\right)\hfill \\ \hfill & =c{m}_{\alpha ,{\alpha }^{-1}\beta }\left({\left(T_1^{\alpha ,{\alpha }^{-1}\beta }\right)}^{-1}(a\otimes b)\right)=c\left(E_{\alpha }\left(a\right)b\right)=c{\epsilon }_{\alpha }\left(a\right)b.\hfill \end{array}$$

□

Lemma 10.

$S_{\alpha \beta }\left(ab\right)=S_{\beta }\left(b\right)S_{\alpha }\left(a\right)$ for all $a\in H_{\alpha }$ and $b\in H_{\beta }$.

Proof. 

We have

$$\begin{array}{cc}& \phantom{=}{m}_{\gamma \alpha \beta ,{\beta }^{-1}{\alpha }^{-1}\delta }\left((H_{\gamma \alpha \beta }\otimes S_{\alpha \beta })\left((c\otimes 1){\Delta }_{\alpha }\left(a\right){\Delta }_{\beta }\left(b\right)\right)(1\otimes d)\right)\hfill \\ & =m_{\gamma \alpha \beta ,{\beta }^{-1}{\alpha }^{-1}\delta }\left((H_{\gamma \alpha \beta }\otimes S_{\alpha \beta })\left((c\otimes 1){\Delta }_{\alpha \beta }\left(ab\right)\right)(1\otimes d)\right)\hfill \\ & =c{\epsilon }_{\alpha \beta }\left(ab\right)d=c{\epsilon }_{\alpha }\left(a\right)d{\epsilon }_{\beta }\left(b\right)=m_{\gamma \alpha ,{\alpha }^{-1}\delta }\left((H_{\gamma \alpha }\otimes S_{\alpha })\left((c\otimes 1){\Delta }_{\alpha }\left(a\right)\right)(1\otimes d)\right){\epsilon }_{\beta }\left(b\right)\hfill \end{array}$$

for all $a\in H_{\alpha },b\in H_{\beta },c\in H_{\gamma }$ and $d\in H_{\delta }$. By the surjectivity of $T_2$, we get

$$\begin{array}{cc}& \phantom{=}{m}_{\gamma \beta ,{\beta }^{-1}{\alpha }^{-1}\delta }\left((H_{\gamma \beta }\otimes S_{\alpha \beta })\left((c\otimes a){\Delta }_{\beta }\left(b\right)\right)(1\otimes d)\right)\hfill \\ & =m_{\gamma ,{\alpha }^{-1}\delta }\left((H_{\gamma }\otimes S_{\alpha })(c\otimes a)(1\otimes d)\right){\epsilon }_{\beta }\left(b\right)=c{S}_{\alpha }\left(a\right)d{\epsilon }_{\beta }\left(b\right)=c{\epsilon }_{\beta }\left(b\right)S_{\alpha }\left(a\right)d\hfill \\ & =m_{\gamma \beta ,{\beta }^{-1}{\alpha }^{-1}\delta }\left((H_{\gamma \beta }\otimes S_{\beta })\left((c\otimes 1){\Delta }_{\beta }\left(b\right)\right)(1\otimes S_{\alpha }\left(a\right)d)\right)\hfill \end{array}$$

for all $a\in H_{\alpha },b\in H_{\beta },c\in H_{\gamma }$ and $d\in H_{\delta }$. Again by the surjectivity of $T_2$, we get

$$m_{\gamma ,{\beta }^{-1}{\alpha }^{-1}\delta }\left((H_{\gamma }\otimes S_{\alpha \beta })(c\otimes ab)(1\otimes d)\right)=m_{\gamma ,{\beta }^{-1}{\alpha }^{-1}\delta }\left((H_{\gamma }\otimes S_{\beta })(c\otimes b)(1\otimes S_{\alpha }\left(a\right)d)\right)$$

whence $c{S}_{\alpha \beta }\left(ab\right)d=c{S}_{\beta }\left(b\right)S_{\alpha }\left(a\right)d.$ □

Define another one family of linear maps $\overline{S}={\{{\overline{S}}_{\alpha }:H_{\alpha }\to H_{{\alpha }^{-1}}\}}_{\alpha \in \pi }$ by

$$a{\overline{S}}_{\beta }\left(b\right)=(H_{\alpha {\beta }^{-1}}\otimes {\epsilon }_{\beta }){\left(T_2^{\alpha {\beta }^{-1},\beta }\right)}^{-1}(a\otimes b)$$

for all $a\in H_{\alpha },b\in H_{\beta }$.

Completely similar as in Lemma 8, we get here that

Lemma 11.

$(c\otimes 1)({\overline{S}}_{\alpha }\otimes H_{\alpha \beta })\left({\Delta }_{\alpha }\left(a\right)(1\otimes b)\right)=\left({\left(T_2^{\gamma {\alpha }^{-1},\alpha }\right)}^{-1}(c\otimes a)\right)(1\otimes b)$.

Lemma 12.

$m_{\gamma {\alpha }^{-1},\alpha \beta }\left((c\otimes 1)(S_{\alpha }\otimes H_{\alpha \beta })\left({\Delta }_{\alpha }\left(a\right)(1\otimes b)\right)\right)=c{\epsilon }_{\alpha }\left(a\right)b.$

Proof. 

By Lemma 11, we get

$$(c\otimes 1)({\overline{S}}_{\alpha }\otimes H_{\alpha \beta })\left({\Delta }_{\alpha }\left(a\right)(1\otimes b)\right)=\left({\left(T_2^{\gamma {\alpha }^{-1},\alpha }\right)}^{-1}(c\otimes a)\right)(1\otimes b).$$

And if we apply $m_{\gamma {\alpha }^{-1},\alpha \beta }$ to the both sides of the above equation, we get the formula in the statement of the lemma with $\overline{S}$ instead of S because

$$m_{\gamma {\alpha }^{-1},\alpha }\left({\left(T_2^{\gamma {\alpha }^{-1},\alpha }\right)}^{-1}(c\otimes a)\right)=c{\epsilon }_{\alpha }\left(a\right).$$

We now show that $S=S^{\prime }$. Indeed, we have, by definition,

$$a{\overline{S}}_{\beta }\left(b\right)=\sum a_i{\epsilon }_{\beta }\left(b_i\right)$$

if $a\otimes b=\sum \left(a_i\otimes 1\right){\Delta }_{\beta }\left(b_i\right)$. If we apply $H_{\alpha }\otimes S_{\beta }$ and multiply with $1\otimes c$ to the both sides of the equation: $a\otimes b=\sum \left(a_i\otimes 1\right){\Delta }_{\beta }\left(b_i\right)$, we get

$$a\otimes S_{\beta }\left(b\right)c=\sum (H_{\alpha }\otimes S_{\beta })\left((a_i\otimes 1){\Delta }_{\beta }\left(b_i\right)\right)(1\otimes c).$$

And if we apply $m_{\alpha ,{\beta }^{-1}\gamma }$ to the both sides of the above equation, we obtain, using Lemma 9, that

$$a{S}_{\beta }\left(b\right)c=\sum a_i{\epsilon }_{\beta }\left(b_i\right)c=a{\overline{S}}_{\beta }\left(b\right)c.$$

This shows that $S_{\beta }\left(b\right)={\overline{S}}_{\beta }\left(b\right)$. This proves the lemma; the formula was already proven for $\overline{S}$. □

Apropos of Lemmas 9 and 12, by setting $\beta =\gamma =1$ and $b=c=1$, we have the usual formulas

$$m_{\alpha ,{\alpha }^{-1}}(H_{\alpha }\otimes S_{\alpha }){\Delta }_{\alpha }\left(a\right)={\epsilon }_{\alpha }\left(a\right)1,m_{{\alpha }^{-1},\alpha }(S_{\alpha }\otimes H_{\alpha }){\Delta }_{\alpha }\left(a\right)={\epsilon }_{\alpha }\left(a\right)1.$$

We have constructed an antihomomorphism $S={\left\{S_{\alpha }\right\}}_{\alpha \in \pi }$ that has the properties of the antipode in the Hopf group-algebra theory.

From the above discussion, we get the following the main result.

Theorem 1.

If $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$ is a unital associative group algebra over $\mathbb{k}$ with a family $\Delta ={\{{\Delta }_{\alpha }:H_{\alpha }⟶H_{\alpha }\otimes H_{\alpha }\}}_{\alpha \in \pi }$ of coassociative comultiplications, then H is a Hopf π-algebra if and only if Δ is a π-algebra homomorphism and the right and left π-Galois maps both have inverses.

4. Hopf (Non)coassociative Group-Algebras

We begin by the main definition of this paper which is slightly dual to the notion of a quasigroup Hopf group-coalgebra studied in [27].

Definition 2.

A Hopf non-coassociative group-algebra over π is a π-algebra $H={(\left\{H_{\alpha }\right\},m={\{m_{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha \beta }\}}_{\alpha ,\beta \in \pi },\eta )}_{\alpha \in \pi }$, endowed with a family $S={\{S_{\alpha }:H_{\alpha }⟶H_{{\alpha }^{-1}}\}}_{\alpha \in \pi }$ of $\mathbb{k}$-linear maps (the antipode) such that the following conditions hold:

$$\begin{array}{cc}\hfill •& \mathit{Each}(H_{\alpha },{\Delta }_{\alpha },{\epsilon }_{\alpha }) with comultiplication{\Delta }_{\alpha }and counit{\epsilon }_{\alpha }\hfill \\ \hfill & is a not necessarily coassociative coalgebra;\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill •for all\alpha ,\beta \in \pi ,& \eta :\mathbb{k}⟶H_{1}and{m}_{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha \beta }\hfill \\ \hfill & are coalgebra homomorphisms;\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill •for\alpha \in \pi ,& (m_{{\alpha }^{-1},\alpha }\otimes i{d}_{H_{\alpha }})(S_{\alpha }\otimes i{d}_{H_{\alpha }}\otimes i{d}_{H_{\alpha }})(i{d}_{H_{\alpha }}\otimes {\Delta }_{\alpha }){\Delta }_{\alpha }\hfill \\ \hfill =\eta \otimes i{d}_{H_{\alpha }}=& (m_{\alpha ,{\alpha }^{-1}}\otimes i{d}_{H_{\alpha }})(i{d}_{H_{\alpha }}\otimes S_{\alpha }\otimes i{d}_{H_{\alpha }})(i{d}_{H_{\alpha }}\otimes {\Delta }_{\alpha }){\Delta }_{\alpha };\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill •for\alpha \in \pi ,& (i{d}_{H_{\alpha }}\otimes m_{{\alpha }^{-1},\alpha })(i{d}_{H_{\alpha }}\otimes S_{\alpha }\otimes i{d}_{H_{\alpha }})({\Delta }_{\alpha }\otimes i{d}_{H_{\alpha }}){\Delta }_{\alpha }\hfill \\ \hfill =i{d}_{H_{\alpha }}\otimes \eta =& (i{d}_{H_{\alpha }}\otimes m_{\alpha ,{\alpha }^{-1}})(i{d}_{H_{\alpha }}\otimes i{d}_{H_{\alpha }}\otimes S_{\alpha })({\Delta }_{\alpha }\otimes i{d}_{H_{\alpha }}){\Delta }_{\alpha }.\hfill \end{array}$$

(16)

We remark that the notion of a Hopf non-coassociative group-algebra is not self-dual and that $(H_1,m_{1,1},\eta ,{\Delta }_1,{\epsilon }_1,S_1)$ is a (classical) Hopf coquasigroup. Let $\pi =\left\{1\right\}$, $H=H_1$ is a (classical) Hopf coquasigroup. One can easily verify that a Hopf non-coassociative group-algebra is a Hopf $\pi$-algebra if and only if its coproduct is coassociative.

In this paper, a Hopf non-coassociative group-algebra over $\pi$ is called Hopf non-coassociative π-algebra.

Remark 3.

(1) 

The axiom (14) amounts to that, for any $\alpha ,\beta \in \pi ,a\in H_{\alpha }$ and $b\in H_{\beta }$,

$$\begin{array}{cc}\hfill {\Delta }_1\left(1\right)& =1\otimes 1,{\epsilon }_1\left(1\right)=1_{\mathbb{k}},\hfill \\ \hfill {\Delta }_{\alpha \beta }\left(ab\right)& ={\Delta }_{\alpha }\left(a\right){\Delta }_{\beta }\left(b\right),{\epsilon }_{\alpha \beta }\left(ab\right)={\epsilon }_{\alpha }\left(a\right){\epsilon }_{\beta }\left(b\right).\hfill \end{array}$$

(2) 

In terms of Sweedler’s notation, the axiom (15) gives that, for any $\alpha \in \pi ,h\in H_{\alpha }$,

$$S_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)\left(1\right)}\otimes h_{\left(2\right)\left(2\right)}=1\otimes h=h_{\left(1\right)}{S}_{\alpha }\left(h_{\left(2\right)\left(1\right)}\right)\otimes h_{\left(2\right)\left(2\right)}.$$

(17)

(3) 

In terms of Sweedler’s notation, the axiom (16) gives that, for any $\alpha \in \pi ,h\in H_{\alpha }$,

$$h_{\left(1\right)\left(1\right)}\otimes S_{\alpha }\left(h_{\left(1\right)\left(2\right)}\right)h_{\left(2\right)}=h\otimes 1=h_{\left(1\right)\left(1\right)}\otimes h_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(h_{\left(2\right)}\right).$$

(18)

Definition 3.

Let H be a Hopf non-coassociative group-algebra. Then, for all $\alpha \in \pi$ and $a\in H_{\alpha }$,

(1) 

H iscommutativeif$m_{\alpha ,{\alpha }^{-1}}=m_{{\alpha }^{-1},\alpha }$.

(2) 

H iscocommutativeif each${\Delta }_{\alpha }$ is cocommutative.

(3) 

H isflexibleif

$$a_{\left(1\right)}{a}_{\left(2\right)\left(2\right)}\otimes a_{\left(2\right)\left(1\right)}=a_{\left(1\right)\left(1\right)}{a}_{\left(2\right)}\otimes a_{\left(1\right)\left(2\right)}.$$

(4) 

H isalternativeif

$$a_{\left(1\right)}{a}_{\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)}=a_{\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}\otimes a_{\left(2\right)},a_{\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}{a}_{\left(2\right)\left(2\right)}=a_{\left(1\right)\left(1\right)}\otimes a_{\left(1\right)\left(2\right)}{a}_{\left(2\right)}.$$

(5) 

H is calledMoufangif

$$a_{\left(1\right)}{a}_{\left(2\right)\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)\left(2\right)}=a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\otimes a_{\left(2\right)}.$$

A Hopf non-coassociative group-algebra H is said to be of finite type if, for all $\alpha \in \pi ,H_{\alpha }$ is finite dimensional (over $\mathbb{k}$). Note that it does not mean that ${\displaystyle \underset{\alpha \in \pi }{⨁}}{H}_{\alpha }$ is finite-dimensional (unless $H_{\alpha }\ne 0$, for all but a finite number of $\alpha \in \pi$).

The antipode $S={\left\{S_{\alpha }\right\}}_{\alpha \in \pi }$ of H is said to be bijective if each $S_{\alpha }$ is bijective. We will later show that it is bijective whenever H is quasitriangular (see Theorem 12).

Example 1.

Let $(H,m,\Delta ,\epsilon ,S)$ be a Hopf coquasigroup and the group π act on H by Hopf coquasigroup endomorphisms.

(1)

Set $H^{\pi }={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$ where the coalgebra $H_{\alpha }$ is a copy of H for each $\alpha \in \pi$. Fix an identification isomorphism of coalgebras $i_{\alpha }:H⟶H_{\alpha }$. For $\alpha ,\beta \in \pi$, one defines a multiplication $m_{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha \beta }$ by

$$m_{\alpha ,\beta }(i_{\alpha }\left(h\right)\otimes i_{\beta }\left(a\right))=\left(i_{\alpha \beta }\left(ha\right)\right)$$

for any $h,a\in H$. The counit ${\epsilon }_1:H_1⟶\mathbb{k}$ is defined by ${\epsilon }_1\left(i_1\left(h\right)\right)=\epsilon \left(h\right)$ for $h\in H$. For any $\alpha \in \pi$, the antipode $S_{\alpha }:H_{\alpha }⟶H_{{\alpha }^{-1}}$ is given by $S_{\alpha }\left(i_{\alpha }\left(h\right)\right)=i_{{\alpha }^{-1}}\left(S\left(h\right)\right)$. All the axioms of a Hopf non-coassociative π-algebra for $H^{\pi }$ follow directly from definitions.

(2)

Let ${\overline{H}}^{\pi }$ be the same family of coalgebras $\{H_{\alpha }=H\}$ with the same counit, the multiplication ${\overline{m}}_{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha \beta }$ and the antipode $S_{\alpha }:H_{\alpha }⟶H_{{\alpha }^{-1}}$ defined by

$$\begin{array}{ccc}& & {\overline{m}}_{\alpha ,\beta }{i}_{\alpha }\left(\beta \left(h\right)\right)\otimes i_{\beta }\left(a\right)=i_{\alpha \beta }\left(h\right)\hfill \\ & & {\overline{S}}_{\alpha }\left(i_{\alpha }\left(h\right)\right)=i_{{\alpha }^{-1}}\left(\alpha \left(S\left(h\right)\right)\right)=i_{{\alpha }^{-1}}(S\left(\alpha \left(h\right)\right)\hfill \end{array}$$

where $h,a\in H$. The axioms of a Hopf non-coassociative π-algebra for ${\overline{H}}^{\pi }$ follow from definitions. Both $H^{\pi }$ and ${\overline{H}}^{\pi }$ are extensions of H since $H_1^{\pi }={\overline{H}}_1^{\pi }=H_1$ as Hopf coquasigroups.

Example 2.

 (1)

Let $A={(\left\{A_{\alpha }\right\},m,\eta )}_{\alpha \in \pi }$ be a π-algebra. Set

$$A_{\alpha }^{op}=A_{{\alpha }^{-1}}and{m}_{\alpha ,\beta }^{op}=m_{{\beta }^{-1},{\alpha }^{-1}}\circ {\sigma }_{A_{{\alpha }^{-1}},A_{{\beta }^{-1}}}.$$

Then $A^{op}={(\left\{A_{\alpha }^{op}\right\},m^{op},\eta )}_{\alpha \in \pi }$ is a π-algebra, called opposite to A.

If $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$ is a Hopf non-coassociative group-algebra whose antipode $S={\left\{S_{\alpha }\right\}}_{\alpha \in \pi }$ is bijective, then the opposite π-algebra $H^{op}$, where $H_{\alpha }^{op}=H_{{\alpha }^{-1}}$ as a coalgebra, is a Hopf non-coassociative π-algebra with antipode $S^{op}={\{S_{\alpha }^{op}=S_{\alpha }^{-1}\}}_{\alpha \in \pi }$.

 (2)

Let $H=({\{H_{\alpha },{\Delta }_{\alpha },{\epsilon }_{\alpha }\}}_{\alpha \in \pi },m,\eta ,S)$ be a Hopf non-coassociative π-algebra. Suppose that the antipode $S={\left\{S_{\alpha }\right\}}_{\alpha \in \pi }$ of H is bijective. For any $\alpha \in \pi$, let $H_{\alpha }^{cop}$ be the coopposite coalgebra to $H_{\alpha }$. Then $H^{cop}={\left\{H_{\alpha }^{cop}\right\}}_{\alpha \in \pi }$, endowed with the multiplication and unit of H and with the antipode $S^{cop}={\{S_{\alpha }^{cop}=S_{{\alpha }^{-1}}^{-1}\}}_{\alpha \in \pi }$, is a Hopf non-coassociative π-algebra called coopposite to H.

 (3)

Let $H=({\{H_{\alpha },{\Delta }_{\alpha },{\epsilon }_{\alpha }\}}_{\alpha \in \pi },m,\eta ,S)$ be a Hopf non-coassociative π-algebra. Even if the antipode of H is not bijective, one can always define a Hopf non-coassociative π-algebra opposite and coopposite to H by setting

$$H_{\alpha }^{op,cop}=H_{{\alpha }^{-1}}^{cop},m_{\alpha ,\beta }^{op,cop}=m_{\alpha ,\beta }^{op},1^{op,cop}=1,and{S}_{\alpha }^{op,cop}=S_{{\alpha }^{-1}}.$$

Definition 4.

Let $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$ and $H^{\prime }={\left\{H_{\alpha }^{\prime }\right\}}_{\alpha \in \pi }$ be Hopf non-coassociative π-algebras. A Hopf non-coassociative π-algebra morphism between H and $H^{\prime }$ is a π-algebra morphism $f={\{f_{\alpha }:H_{\alpha }⟶H_{\alpha }^{\prime }\}}_{\alpha \in \pi }$ between H and $H^{\prime }$ such that, for any $\alpha \in \pi ,$$f_{\alpha }$ is a coalgebra morphism and $f_{{\alpha }^{-1}}\circ S_{\alpha }=S_{\alpha }^{\prime }\circ f_{\alpha }$. The Hopf non-coassociative π-algebra isomorphism $f={\{f_{\alpha }:H_{\alpha }⟶H_{\alpha }^{\prime }\}}_{\alpha \in \pi }$ is a Hopf non-coassociative π-algebra morphism in which each $f_{\alpha }$ is a linear isomorphism.

Let us first remark that, when $\pi$ is a finite group, there is a one-to-one correspondence between (isomorphic classes of) $\pi$-algebras and (isomorphic classes of) $\pi$-graded algebras. Recall that an algebra $(A,m,\eta )$ is $\pi$-graded if A admits a decomposition as a direct sum of $\mathbb{k}$-spaces $A={\displaystyle \underset{\alpha \in \pi }{⨁}}{A}_{\alpha }$ such that

$$\begin{array}{cc}\hfill A_{\alpha }{A}_{\beta }& \subset A_{\alpha \beta },\forall \alpha ,\beta \in \pi .\hfill \\ \hfill 1& \in A_1.\hfill \end{array}$$

Let us denote by ${\pi }_{\alpha }:A_{\alpha }⟶A$ the canonical injection. Then ${\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ is a $\pi$-algebra with multiplication $\left\{m({\pi }_{\alpha }\otimes {\pi }_{\beta })\right|A_{\alpha }\otimes A_{\beta }\}$ and unit $\eta$. Conversely, if $A={(\left\{A_{\alpha }\right\},m,\eta )}_{\alpha \in \pi }$ is a $\pi$-algebra, then $\tilde{A}={\displaystyle \underset{\alpha \in \pi }{⨁}}{A}_{\alpha }$ is a $\pi$-graded algebra with multiplication $\tilde{m}$ and unit $\tilde{\eta }$ given on the summands by

$$\tilde{m}|A_{\alpha }\otimes A_{\beta }=m_{\alpha ,\beta }and\tilde{\eta }=\eta .$$

Let now $H={(\{H_{\alpha },{\Delta }_{\alpha },{\epsilon }_{\alpha }\},m,1,S)}_{\alpha \in \pi }$ be a Hopf non-coassociative group-algebra, where $\pi$ is a finite group. Then the algebra $(\tilde{H},\tilde{m},\tilde{\eta }),$ defined as above, is a Hopf coquasigroup with comultiplication $\tilde{\Delta }$, counit element $\tilde{\epsilon }$, and antipode $\tilde{S}$ given by

$$\tilde{\Delta }|H_{\alpha }={\Delta }_{\alpha }\tilde{\epsilon }=\sum _{\alpha \in \pi }{\epsilon }_{\alpha },\tilde{S}=\sum _{\alpha \in \pi }{S}_{\alpha }.$$

In what follows, we study structure properties for a Hopf non-coassociative $\pi$-algebra.

Theorem 2.

Let $H=({\{H_{\alpha },{\Delta }_{\alpha },{\epsilon }_{\alpha }\}}_{\alpha \in \pi },m,\eta ,S)$ be a Hopf non-coassociative π-algebra. Then

$$\begin{array}{cc}\hfill m_{{\alpha }^{-1},\alpha }(S_{\alpha }\otimes i{d}_{H_{\alpha }}){\Delta }_{\alpha }& =1{\epsilon }_{\alpha }=m_{\alpha ,{\alpha }^{-1}}(i{d}_{H_{\alpha }}\otimes S_{\alpha }){\Delta }_{\alpha },\forall \alpha \in \pi .\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill S_{\alpha \beta }\left(ab\right)& =S_{\beta }\left(b\right)S_{\alpha }\left(a\right),\forall \alpha ,\beta \in \pi ,a\in H_{\alpha },b\in H_{\beta };\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill S_1\left(1\right)& =1;\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\Delta }_{{\alpha }^{-1}}{S}_{\alpha }& ={\sigma }_{H_{{\alpha }^{-1}},H_{{\alpha }^{-1}}}(S_{\alpha }\otimes S_{\alpha }){\Delta }_{\alpha },\forall \alpha \in \pi ;\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\epsilon }_{{\alpha }^{-1}}{S}_{\alpha }& ={\epsilon }_{\alpha },\forall \alpha \in \pi .\hfill \end{array}$$

(23)

Proof. 

Equation (19) is directly obtained by applying $i{d}_{H_1}\otimes {\epsilon }_{\alpha }$ to Equation (17) in the definition of a Hopf nonassociative $\pi$-coalgebra. We now show Equation (20) as follows:

$$\begin{array}{cc}\hfill & S_{\alpha \beta }\left(ab\right)=S_{\alpha \beta }{m}_{\alpha ,\beta }(a\otimes b)=S_{\alpha \beta }{m}_{\alpha ,\beta }{\sigma }_{H_{\beta },H_{\alpha }}(b\otimes a)=1S_{\alpha \beta }{m}_{\alpha ,\beta }{\sigma }_{H_{\beta },H_{\alpha }}(b\otimes a)\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(1\otimes S_{\alpha \beta }{m}_{\alpha ,\beta }{\sigma }_{H_{\beta },H_{\alpha }}(b\otimes a))\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(i{d}_{H_1}\otimes S_{\alpha \beta }{m}_{\alpha ,\beta }{\sigma }_{H_{\beta },H_{\alpha }})(1\otimes b\otimes a)\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(i{d}_{H_1}\otimes S_{\alpha \beta }{m}_{\alpha ,\beta }{\sigma }_{H_{\beta },H_{\alpha }})(S_{\beta }\left(b_{\left(1\right)}\right)b_{\left(2\right)\left(1\right)}\otimes b_{\left(2\right)\left(2\right)}\otimes a)\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(S_{\beta }\left(b_{\left(1\right)}\right)b_{\left(2\right)\left(1\right)}\otimes S_{\alpha \beta }{m}_{\alpha ,\beta }{\sigma }_{H_{\beta },H_{\alpha }}(b_{\left(2\right)\left(2\right)}\otimes a))\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(S_{\beta }\left(b_{\left(1\right)}\right)b_{\left(2\right)\left(1\right)}\otimes S_{\alpha \beta }{m}_{\alpha ,\beta }(a\otimes b_{\left(2\right)\left(2\right)}))\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(S_{\beta }\left(b_{\left(1\right)}\right)b_{\left(2\right)\left(1\right)}\otimes S_{\alpha \beta }\left(a{b}_{\left(2\right)\left(2\right)}\right))\hfill \\ \hfill =& \left(S_{\beta }\left(b_{\left(1\right)}\right)b_{\left(2\right)\left(1\right)}\right)S_{\alpha \beta }\left(a{b}_{\left(2\right)\left(2\right)}\right)=\left(S_{\beta }\left(b_{\left(1\right)}\right)1b_{\left(2\right)\left(1\right)}\right)S_{\alpha \beta }\left(a{b}_{\left(2\right)\left(2\right)}\right)\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(m_{{\beta }^{-1},1,\beta }(S_{\beta }\left(b_{\left(1\right)}\right)\otimes 1\otimes b_{\left(2\right)\left(1\right)})\otimes S_{\alpha \beta }{m}_{\alpha ,\beta }(a\otimes b_{\left(2\right)\left(2\right)}))\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(m_{{\beta }^{-1},1,\beta }\otimes S_{\alpha \beta }{m}_{\alpha ,\beta })(S_{\beta }\left(b_{\left(1\right)}\right)\otimes 1\otimes b_{\left(2\right)\left(1\right)}\otimes a\otimes b_{\left(2\right)\left(2\right)}))\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(m_{{\beta }^{-1},1,\beta }\otimes S_{\alpha \beta }{m}_{\alpha ,\beta })(i{d}_{H_{{\beta }^{-1}}}\otimes i{d}_{H_1}\otimes {\sigma }_{H_{\alpha },H_{\beta }}\otimes i{d}_{H_{\beta }})\hfill \\ \hfill & (S_{\beta }\left(b_{\left(1\right)}\right)\otimes 1\otimes a\otimes b_{\left(2\right)\left(1\right)}\otimes b_{\left(2\right)\left(2\right)}))\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(m_{{\beta }^{-1},1,\beta }\otimes S_{\alpha \beta }{m}_{\alpha ,\beta })(i{d}_{H_{{\beta }^{-1}}}\otimes i{d}_{H_1}\otimes {\sigma }_{H_{\alpha },H_{\beta }}\otimes i{d}_{H_{\beta }})\hfill \\ \hfill & (S_{\beta }\left(b_{\left(1\right)}\right)\otimes S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)}\otimes b_{\left(2\right)\left(1\right)}\otimes b_{\left(2\right)\left(2\right)}))\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(m_{{\beta }^{-1},1,\beta }\otimes S_{\alpha \beta }{m}_{\alpha ,\beta })(S_{\beta }\left(b_{\left(1\right)}\right)\otimes S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)}\otimes b_{\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)}\otimes b_{\left(2\right)\left(2\right)}))\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(m_{{\beta }^{-1},1,\beta }(S_{\beta }\left(b_{\left(1\right)}\right)\otimes S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)}\otimes b_{\left(2\right)\left(1\right)})\otimes S_{\alpha \beta }{m}_{\alpha ,\beta }(a_{\left(2\right)\left(2\right)}\otimes b_{\left(2\right)\left(2\right)})))\hfill \\ \hfill =& m_{1,{\beta }^{-1}{\alpha }^{-1}}(S_{\beta }\left(b_{\left(1\right)}\right)\left(S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)}\right)b_{\left(2\right)\left(1\right)}\otimes S_{\alpha \beta }\left(a_{\left(2\right)\left(2\right)}{b}_{\left(2\right)\left(2\right)}\right)))\hfill \\ \hfill =& \left(S_{\beta }\left(b_{\left(1\right)}\right)\left(S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)}\right)b_{\left(2\right)\left(1\right)}\right)S_{\alpha \beta }\left(a_{\left(2\right)\left(2\right)}{b}_{\left(2\right)\left(2\right)}\right)\hfill \\ \hfill =& S_{\beta }\left(b_{\left(1\right)}\right)S_{\alpha }\left(a_{\left(1\right)}\right)\left(a_{\left(2\right)\left(1\right)}{b}_{\left(2\right)\left(1\right)}{S}_{\alpha \beta }\left(a_{\left(2\right)\left(2\right)}{b}_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ \hfill =& S_{\beta }\left(b_{\left(1\right)}\right)S_{\alpha }\left(a_{\left(1\right)}\right)\left({\left(a_{\left(2\right)}{b}_{\left(2\right)}\right)}_{\left(1\right)}{S}_{\alpha \beta }\left({\left(a_{\left(2\right)}{b}_{\left(2\right)}\right)}_{\left(2\right)}\right)\right)\hfill \\ \hfill =& S_{\beta }\left(b_{\left(1\right)}\right)S_{\alpha }\left(a_{\left(1\right)}\right)\epsilon \left(a_{\left(2\right)}{b}_{\left(2\right)}\right)=S_{\beta }\left(b_{\left(1\right)}\right)S_{\alpha }\left(a_{\left(1\right)}\right)\epsilon \left(a_{\left(2\right)}\right)\epsilon \left(b_{\left(2\right)}\right)\hfill \\ \hfill =& S_{\beta }\left(b_{\left(1\right)}\epsilon \left(b_{\left(2\right)}\right)\right)S_{\alpha }\left(a_{\left(1\right)}\epsilon \left(a_{\left(2\right)}\right)\right)=S_{\beta }\left(b\right)S_{\alpha }\left(a\right).\hfill \end{array}$$

Thus, $S_{\alpha \beta }\left(ab\right)=S_{\beta }\left(b\right)S_{\alpha }\left(a\right),\forall \alpha ,\beta \in \pi ,a\in H_{\alpha },b\in H_{\beta }.$

To show Equation (22), for all, $\alpha \in \pi ,h\in H_{\alpha }$, we have that

$$\begin{array}{cc}\hfill & (S_{\alpha }\otimes S_{\alpha }){\Delta }_{\alpha }\left(h\right)=S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)}\right)\hfill \\ \hfill =& (m_{{\alpha }^{-1},1}\otimes m_{{\alpha }^{-1},1})(i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_{{\alpha }^{-1}},H_1}\otimes i{d}_{H_1})(i{d}_{H_{{\alpha }^{-1}}}\otimes i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_1,H_1})\hfill \\ \hfill & (S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)}\right)\otimes 1\otimes 1)\hfill \\ \hfill =& (m_{{\alpha }^{-1},1}\otimes m_{{\alpha }^{-1},1})(i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_{{\alpha }^{-1}},H_1}\otimes i{d}_{H_1})(i{d}_{H_{{\alpha }^{-1}}}\otimes i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_1,H_1})\hfill \\ \hfill & (S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)}\right)\otimes \Delta \left(1\right))\hfill \\ \hfill =& (m_{{\alpha }^{-1},1}\otimes m_{{\alpha }^{-1},1})(i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_{{\alpha }^{-1}},H_1}\otimes i{d}_{H_1})(i{d}_{H_{{\alpha }^{-1}}}\otimes i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_1,H_1})\hfill \\ \hfill & (S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right)\otimes \Delta \left(h_{\left(2\right)\left(1\right)\left(2\right)}{S}_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right)\right))\hfill \\ \hfill =& (m_{{\alpha }^{-1},1}\otimes m_{{\alpha }^{-1},1})(i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_{{\alpha }^{-1}},H_1}\otimes i{d}_{H_1})(i{d}_{H_{{\alpha }^{-1}}}\otimes i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_1,H_1})\hfill \\ \hfill & (S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right)\otimes \Delta \left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)\Delta \left(S_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right)\right))\hfill \\ \hfill =& (m_{{\alpha }^{-1},1}\otimes m_{{\alpha }^{-1},1})(i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_{{\alpha }^{-1}},H_1}\otimes i{d}_{H_1})(i{d}_{H_{{\alpha }^{-1}}}\otimes i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_1,H_1})\hfill \\ \hfill & (S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right)\otimes \Delta \left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)\Delta S_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right))\hfill \\ \hfill =& (m_{{\alpha }^{-1},1}\otimes m_{{\alpha }^{-1},1})(i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_{{\alpha }^{-1}},H_1}\otimes i{d}_{H_1})(i{d}_{H_{{\alpha }^{-1}}}\otimes i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_1,H_1})\hfill \\ \hfill & (S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right)\otimes (h_{\left(2\right)\left(1\right)\left(2\right)\left(1\right)}\otimes h_{\left(2\right)\left(1\right)\left(2\right)\left(2\right)})(S_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(1\right)}\otimes S_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(2\right)}))\hfill \\ \hfill =& (m_{{\alpha }^{-1},1}\otimes m_{{\alpha }^{-1},1})(i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_{{\alpha }^{-1}},H_1}\otimes i{d}_{H_1})(i{d}_{H_{{\alpha }^{-1}}}\otimes i{d}_{H_{{\alpha }^{-1}}}\otimes {\sigma }_{H_1,H_1})\hfill \\ \hfill & (S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right)\otimes h_{\left(2\right)\left(1\right)\left(2\right)\left(1\right)}{S}_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(1\right)}\otimes h_{\left(2\right)\left(1\right)\left(2\right)\left(2\right)}{S}_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(2\right)})\hfill \\ \hfill =& S_{\alpha }\left(h_{\left(1\right)}\right)\left(h_{\left(2\right)\left(1\right)\left(2\right)\left(2\right)}{S}_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(2\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right)\left(h_{\left(2\right)\left(1\right)\left(2\right)\left(1\right)}{S}_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(1\right)}\right)\hfill \\ \hfill =& S_{\alpha }\left(h_{\left(1\right)}\right)\left(h_{\left(2\right)\left(1\right)\left(2\right)\left(2\right)}{S}_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(2\right)}\right)\otimes \left(S_{\alpha }\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right)h_{\left(2\right)\left(1\right)\left(2\right)\left(1\right)}\right)S_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(1\right)}\hfill \\ \hfill =& S_{\alpha }\left(h_{\left(1\right)}\right)\left(h_{\left(2\right)\left(1\right)}{S}_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(2\right)}\right)\otimes S_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(1\right)}=\left(S_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)\left(1\right)}\right)S_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(2\right)}\otimes S_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(1\right)}\hfill \\ \hfill =& (S_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)\left(1\right)}\otimes 1)(S_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(2\right)}\otimes S_{\alpha }{\left(h_{\left(2\right)\left(2\right)}\right)}_{\left(1\right)})\hfill \\ \hfill =& (S_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)\left(1\right)}\otimes 1){\sigma }_{H_{{\alpha }^{-1}},H_{{\alpha }^{-1}}}{\Delta }_{{\alpha }^{-1}}{S}_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right)\hfill \\ \hfill =& (1\otimes 1){\sigma }_{H_{{\alpha }^{-1}},H_{{\alpha }^{-1}}}{\Delta }_{{\alpha }^{-1}}{S}_{\alpha }\left(h\right)={\sigma }_{H_{{\alpha }^{-1}},H_{{\alpha }^{-1}}}{\Delta }_{{\alpha }^{-1}}{S}_{\alpha }\left(h\right).\hfill \end{array}$$

Thus, ${\Delta }_{{\alpha }^{-1}}{S}_{\alpha }={\sigma }_{H_{{\alpha }^{-1}},H_{{\alpha }^{-1}}}(S_{\alpha }\otimes S_{\alpha }){\Delta }_{\alpha },\forall \alpha \in \pi .$

Using Equation (19), we obtain Equation (21):

$$m_{1,1}(S_1\otimes i{d}_{H_1}){\Delta }_1\left(1\right)=1{\epsilon }_1\left(1\right)⟹S_1\left(1\right)1=1⟹S_1\left(1\right)=1.$$

We can obtain Equation (23) also by Equation (19): $\forall \alpha \in \pi ,h\in H_{\alpha },$

$$\begin{array}{cc}\hfill & m_{{\alpha }^{-1},\alpha }(S_{\alpha }\otimes i{d}_{H_{\alpha }}){\Delta }_{\alpha }\left(h\right)=1{\epsilon }_{\alpha }\left(h\right)⟹S_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)}=1{\epsilon }_{\alpha }\left(h\right)\hfill \\ \hfill ⟹& {\epsilon }_1\left(S_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)}\right)={\epsilon }_1\left(1{\epsilon }_{\alpha }\left(h\right)\right)⟹{\epsilon }_{{\alpha }^{-1}}\left(S_{\alpha }\left(h_{\left(1\right)}\right)\right){\epsilon }_{\alpha }\left(h_{\left(2\right)}\right)={\epsilon }_1\left(1\right){\epsilon }_{\alpha }\left(h\right)\hfill \\ \hfill ⟹& {\epsilon }_{{\alpha }^{-1}}\left(S_{\alpha }\left(h_{\left(1\right)}{\epsilon }_{\alpha }\left(h_{\left(2\right)}\right)\right)\right)=1_{\mathbb{k}}{\epsilon }_{\alpha }\left(h\right)⟹{\epsilon }_{{\alpha }^{-1}}\left(S_{\alpha }\left(h\right)\right)={\epsilon }_{\alpha }\left(h\right),\hfill \end{array}$$

i.e., ${\epsilon }_{{\alpha }^{-1}}{S}_{\alpha }={\epsilon }_{\alpha },\forall \alpha \in \pi .$ □

Corollary 1.

The antipode of a Hopf non-coassociative π-algebra is unique.

Proof. 

If $S,\widehat{S}$ are two antipodes on a Hopf non-coassociative $\pi$-algebra H, then they are equal in that, for any $\alpha \in \pi$ and $h\in H_{\alpha }$,

$$\begin{array}{cc}\hfill {\widehat{S}}_{\alpha }\left(h\right)& ={\widehat{S}}_{\alpha }\left(h_{\left(1\right)}{\epsilon }_{\alpha }\left(h_{\left(2\right)}\right)\right)={\widehat{S}}_{\alpha }\left(h_{\left(1\right)}\right){\epsilon }_{\alpha }\left(h_{\left(2\right)}\right)1={\widehat{S}}_{\alpha }\left(h_{\left(1\right)}\right)\left(h_{\left(2\right)\left(1\right)}{S}_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ \hfill =& \left({\widehat{S}}_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)\left(1\right)}\right)S_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right)=1S_{\alpha }\left(h\right)=S_{\alpha }\left(h\right).\hfill \end{array}$$

□

Corollary 2.

Let $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$ be a Hopf non-coassociative π-algebra with the antipode $S={\left\{S_{\alpha }\right\}}_{\alpha \in \pi }$. Then $S_{\alpha }$ is the unique convolution inverse of $i{d}_{H_{\alpha }}$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$, for all $\alpha \in \pi$.

Proof. 

Equation (19) says that $S_{\alpha }$ is a convolution inverse of $i{d}_{H_{\alpha }}$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$, for all $\alpha \in \pi$. Fix $\alpha \in \pi$. Let $T_{\alpha }$ be a right convolution inverse of $i{d}_{H_{\alpha }}$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$. For all $h\in H_{\alpha }$, we compute

$$\begin{array}{cc}\hfill S_{\alpha }\left(h\right)=& S_{\alpha }\ast (i{d}_{H_{\alpha }}\ast T_{\alpha })\left(h\right)=S_{\alpha }\left(h_{\left(1\right)}\right)(i{d}_{H_{\alpha }}\ast T_{\alpha })\left(h_{\left(2\right)}\right)=S_{\alpha }\left(h_{\left(1\right)}\right)\left(h_{\left(2\right)\left(1\right)}{T}_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ \hfill =& \left(S_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)\left(1\right)}\right)T_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right) \stackrel{\mathrm{by} \mathrm{Equation} \left(17\right)}{=} 1T_{\alpha }\left(h\right)=T_{\alpha }\left(h\right)\hfill \\ \hfill ⟹T_{\alpha }=& S_{\alpha }.\hfill \end{array}$$

Fix $\alpha \in \pi$. Let $T_{\alpha }$ now be a left convolution inverse of $i{d}_{H_{\alpha }}$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$. Similarly, we have $T_{\alpha }=S_{\alpha }$. Therefore, $S_{\alpha }$ is the unique convolution inverse of $i{d}_{H_{\alpha }}$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$, for all $\alpha \in \pi$. □

Similarly, one can get

Corollary 3.

Let $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$ be a Hopf non-coassociative π-algebra with the antipode $S={\left\{S_{\alpha }\right\}}_{\alpha \in \pi }$. Then $i{d}_{H_{\alpha }}$ is the unique convolution inverse of $S_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$, for all $\alpha \in \pi$.

Corollary 4.

Let $H=({\{H_{\alpha },{\Delta }_{\alpha },{\epsilon }_{\alpha }\}}_{\alpha \in \pi },m,\eta ,S)$ be a Hopf non-coassociative π-algebra. Then $\{\alpha \in \pi ,|H_{\alpha }\ne 0\}$ is a subgroup of π.

Proof. 

Set $G=\{\alpha \in \pi ,|H_{\alpha }\ne 0\}$. Since ${\epsilon }_1\left(1\right)=1_{\mathbb{k}}\ne 0$, we first have $0\ne 1\in H_1$, i.e., $H_1\ne 0$, and so $1\in G$.

Now let $\alpha \in G$ whereby $H_{\alpha }\ne 0$, then there exists $0\ne a\in H_{\alpha }.$ Using Equation (13), one can see that $a_{\left(1\right)}{\epsilon }_{\alpha }\left(a_{\left(2\right)}\right)={\epsilon }_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)}=a\ne 0$. It follows that $\exists h\in H_{\alpha },s.t.{\epsilon }_{\alpha }\left(h\right)\ne 0$. Then let $\beta \in G$. In a similar manner, one can also obtain that $\exists g\in H_{\beta },s.t.{\epsilon }_{\beta }\left(g\right)\ne 0$. Thus, ${\epsilon }_{\alpha \beta }\left(hg\right)={\epsilon }_{\alpha }\left(h\right){\epsilon }_{\beta }\left(g\right)\ne 0$, i.e., $0\ne hg\in H_{\alpha \beta }$ and so $\alpha \beta \in G$.

Finally, let $\alpha \in G$. By Equation (23), ${\epsilon }_{{\alpha }^{-1}}{S}_{\alpha }\left(h\right)={\epsilon }_{\alpha }\left(h\right)\ne 0$. Therefore $0\ne S_{\alpha }\left(h\right)\in H_{{\alpha }^{-1}}$ and hence ${\alpha }^{-1}\in G$. □

The following theorem sheds considerable light on the concept of a Hopf non-coassociative $\pi$-algebra morphism.

Theorem 3.

Let $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$ and $H^{\prime }={\left\{H_{\alpha }^{\prime }\right\}}_{\alpha \in \pi }$ be Hopf non-coassociative π-algebras. A π-algebra morphism $f={\{f_{\alpha }:H_{\alpha }⟶H_{\alpha }^{\prime }\}}_{\alpha \in \pi }$ between H and $H^{\prime }$ such that, for any $\alpha \in \pi ,$$f_{\alpha }$ is a coalgebra morphism satisfies $f_{{\alpha }^{-1}}\circ S_{\alpha }=S_{\alpha }^{\prime }\circ f_{\alpha }$, for all $\alpha \in \pi$.

Proof. 

Consider the convolution inverse of $f_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H^{\prime })$,

$$\begin{array}{cc}\hfill S_{\alpha }^{\prime }\circ f_{\alpha }\ast f_{\alpha }\left(h\right)=& S_{\alpha }^{\prime }\circ f_{\alpha }\left(h_{\left(1\right)}\right)f_{\alpha }\left(h_{\left(2\right)}\right)=S_{\alpha }^{\prime }\left({\left(f_{\alpha }\left(h\right)\right)}_{\left(1\right)}\right){\left(f_{\alpha }\left(h\right)\right)}_{\left(2\right)}={\epsilon }_{\alpha }^{\prime }\left(f_{\alpha }\left(h\right)\right)1^{\prime }={\epsilon }_{\alpha }\left(h\right)1^{\prime },\hfill \end{array}$$

whence $S_{\alpha }^{\prime }\circ f_{\alpha }$ is a left convolution inverse of $f_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H^{\prime })$,

$$\begin{array}{cc}\hfill f_{\alpha }\ast f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)=& f_{\alpha }\left(h_{\left(1\right)}\right)f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h_{\left(2\right)}\right)=f_1\left(h_{\left(1\right)}{S}_{\alpha }\left(h_{\left(2\right)}\right)\right)=f_1\left({\epsilon }_{\alpha }\left(h\right)1\right)={\epsilon }_{\alpha }\left(h\right)1^{\prime },\hfill \end{array}$$

whence $f_{{\alpha }^{-1}}\circ S_{\alpha }$ is a right convolution inverse of $f_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H^{\prime })$,

$$\begin{array}{cc}\hfill & f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)=(S_{\alpha }^{\prime }\circ f_{\alpha }\ast f_{\alpha })\ast f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)=S_{\alpha }^{\prime }\circ f_{\alpha }\ast f_{\alpha }\left(h_{\left(1\right)}\right)f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h_{\left(2\right)}\right)\hfill \\ \hfill =& \left(S_{\alpha }^{\prime }\circ f_{\alpha }\left(h_{\left(1\right)\left(1\right)}\right)f_{\alpha }\left(h_{\left(1\right)\left(2\right)}\right)\right)f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h_{\left(2\right)}\right)=S_{\alpha }^{\prime }\circ f_{\alpha }\left(h_{\left(1\right)\left(1\right)}\right)\left(f_{\alpha }\left(h_{\left(1\right)\left(2\right)}\right)f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h_{\left(2\right)}\right)\right)\hfill \\ \hfill =& S_{\alpha }^{\prime }\circ f_{\alpha }\left(h_{\left(1\right)\left(1\right)}\right)f_1\left(h_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(h_{\left(2\right)}\right)\right)=S_{\alpha }^{\prime }\circ f_{\alpha }\left(h\right)f_1\left(1\right)=S_{\alpha }^{\prime }\circ f_{\alpha }\left(h\right)1^{\prime }=S_{\alpha }^{\prime }\circ f_{\alpha }\left(h\right),\hfill \end{array}$$

from which we obtain $f_{{\alpha }^{-1}}\circ S_{\alpha }=S_{\alpha }^{\prime }\circ f_{\alpha }$. This completes the proof. □

By looking into the proof of Theorem 3, we note that $f_{{\alpha }^{-1}}\circ S_{\alpha }=S_{\alpha }^{\prime }\circ f_{\alpha }$ and $f_{\alpha }$ are convolution inverses in the convolution algebra $\mathrm{Conv}(H_{\alpha },H^{\prime })$. More precisely, we claim:

Corollary 5.

If $f={\{f_{\alpha }:H_{\alpha }⟶H_{\alpha }^{\prime }\}}_{\alpha \in \pi }$ is a Hopf non-coassociative π-algebra morphism between H and $H^{\prime }$. Then:

 (1)

$f_{{\alpha }^{-1}}\circ S_{\alpha }=S_{\alpha }^{\prime }\circ f_{\alpha }$ is the unique convolution inverse of $f_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H^{\prime })$;

 (2)

$f_{\alpha }$ is the unique convolution inverse of $f_{{\alpha }^{-1}}\circ S_{\alpha }=S_{\alpha }^{\prime }\circ f_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H^{\prime })$.

Proof. 

We first establish part (1). Fix $\alpha \in \pi$. Let $T_{\alpha }$ be a right convolution inverse of $f_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H^{\prime })$.

$$\begin{array}{cc}\hfill f_{{\alpha }^{-1}}& \circ S_{\alpha }\left(h\right)=f_{{\alpha }^{-1}}\circ S_{\alpha }\ast (f_{\alpha }\ast T_{\alpha })\left(h\right)=f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h_{\left(1\right)}\right)f_{\alpha }\ast T_{\alpha }\left(h_{\left(2\right)}\right)\hfill \\ \hfill =& f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h_{\left(1\right)}\right)\left(f_{\alpha }\left(h_{\left(2\right)\left(1\right)}\right)T_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right)\right)=\left(f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h_{\left(1\right)}\right)f_{\alpha }\left(h_{\left(2\right)\left(1\right)}\right)\right)T_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right)\hfill \\ \hfill =& f_1\left(S_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)\left(1\right)}\right)T_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right)=f_1\left(1\right)T_{\alpha }\left(h\right)=1^{\prime }{T}_{\alpha }\left(h\right)=T_{\alpha }\left(h\right).\hfill \end{array}$$

Fix $\alpha \in \pi$. Let $T_{\alpha }$ now be a left convolution inverse of $f_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H^{\prime })$. Similarly, we have $f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)=T_{\alpha }\left(h\right)$.

$f_{{\alpha }^{-1}}\circ S_{\alpha }=S_{\alpha }^{\prime }\circ f_{\alpha }$ is therefore the unique convolution inverse of $f_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H^{\prime })$, for all $\alpha \in \pi$.

We now turn to part (2). Fix $\alpha \in \pi$. Let $T_{\alpha }$ be a right convolution inverse of $f_{{\alpha }^{-1}}\circ S_{\alpha }=S_{\alpha }^{\prime }\circ f_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H^{\prime })$.

$$\begin{array}{cc}\hfill f_{\alpha }\left(h\right)=& f_{\alpha }\ast (f_{{\alpha }^{-1}}\circ S_{\alpha }\ast T_{\alpha })\left(h\right)=f_{\alpha }\left(h_{\left(1\right)}\right)f_{{\alpha }^{-1}}\circ S_{\alpha }\ast T_{\alpha }\left(h_{\left(2\right)}\right)\hfill \\ \hfill =& f_{\alpha }\left(h_{\left(1\right)}\right)\left(f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h_{\left(2\right)\left(1\right)}\right)T_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right)\right)=\left(f_{\alpha }\left(h_{\left(1\right)}\right)f_{{\alpha }^{-1}}\circ S_{\alpha }\left(h_{\left(2\right)\left(1\right)}\right)\right)T_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right)\hfill \\ \hfill =& f_1\left(h_{\left(1\right)}{S}_{\alpha }\left(h_{\left(2\right)\left(1\right)}\right)\right)T_{\alpha }\left(h_{\left(2\right)\left(2\right)}\right)=f_1\left(1\right)T_{\alpha }\left(h\right)=1^{\prime }{T}_{\alpha }\left(h\right)=T_{\alpha }\left(h\right).\hfill \end{array}$$

Fix $\alpha \in \pi$. Let $T_{\alpha }$ now be a left convolution inverse of $f_{{\alpha }^{-1}}\circ S_{\alpha }=S_{\alpha }^{\prime }\circ f_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H^{\prime })$. Similarly, we have $f_{\alpha }\left(h\right)=T_{\alpha }\left(h\right)$. Therefore, $f_{\alpha }$ is the unique convolution inverse of $f_{{\alpha }^{-1}}\circ S_{\alpha }=S_{\alpha }^{\prime }\circ f_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H^{\prime })$, for all $\alpha \in \pi$. □

The following two corollaries can be directly deduced from Theorems 2 and 3.

Corollary 6.

If H is a Hopf non-coassociative π-algebra, then the map $S:H⟶H^{op,cop}$ (where both are opposite and $S^{op,cop}={\{S_{\alpha }^{op,cop}=S_{{\alpha }^{-1}}\}}_{\alpha \in \pi }$) is a Hopf non-coassociative π-algebra isomorphism.

Corollary 7.

If H is a Hopf non-coassociative π-algebra with an invertible antipode S, then the map $S:H⟶H^{op,cop}$ (where both are opposite and $S^{op,cop}={\{S_{\alpha }^{op,cop}=S_{{\alpha }^{-1}}\}}_{\alpha \in \pi }$) is a Hopf non-coassociative π-algebra isomorphism.

Theorem 4.

Let H be a Hopf non-coassociative π-algebra. Then for any $\alpha \in \pi ,$$S_{{\alpha }^{-1}}{S}_{\alpha }=i{d}_{H_{\alpha }}$ if H is commutative or cocommutative.

Proof. 

For any $\alpha \in \pi$. Let $h\in H_{\alpha }$. If H is commutative, we have

$$\begin{array}{cc}\hfill & S_{{\alpha }^{-1}}{S}_{\alpha }\left(h\right)=S_{{\alpha }^{-1}}{S}_{\alpha }\left(h_{\left(1\right)}\epsilon \left(h_{\left(2\right)}\right)\right)=S_{{\alpha }^{-1}}{S}_{\alpha }\left(h_{\left(1\right)}\right)\epsilon \left(h_{\left(2\right)}\right)\hfill \\ \hfill & =S_{{\alpha }^{-1}}{S}_{\alpha }\left(h_{\left(1\right)}\right)\left(S_{\alpha }\left(h_{\left(2\right)\left(1\right)}\right)h_{\left(2\right)\left(2\right)}\right)=\left(S_{{\alpha }^{-1}}{S}_{\alpha }\left(h_{\left(1\right)}\right)S_{\alpha }\left(h_{\left(2\right)\left(1\right)}\right)\right)h_{\left(2\right)\left(2\right)}\hfill \\ \hfill & =S_1\left(h_{\left(2\right)\left(1\right)}{S}_{\alpha }\left(h_{\left(1\right)}\right)\right)h_{\left(2\right)\left(2\right)}=S_1\left(S_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)\left(1\right)}\right)h_{\left(2\right)\left(2\right)}=S_1\left(1\right)h=1h=h.\hfill \end{array}$$

It follows that $S_{{\alpha }^{-1}}{S}_{\alpha }=i{d}_{H_{\alpha }}.$

Similar to the case of H being cocommutative. □

Theorem 5.

Let H be a Hopf non-coassociative π-algebra such that each $S_{\alpha }^{-1}$ exists, for all $\alpha \in \pi$. Then the following identities are equivalent:

 (1)

$a_{\left(2\right)\left(1\right)}{S}_{\alpha }\left(a_{\left(1\right)}\right)\otimes a_{\left(2\right)\left(2\right)}=a_{\left(2\right)}{S}_{\alpha }\left(a_{\left(1\right)\left(2\right)}\right)\otimes a_{\left(1\right)\left(1\right)}=1\otimes a$, for all $\alpha \in \pi ,a\in H_{\alpha }$.

 (2)

$a_{\left(2\right)\left(2\right)}\otimes S_{\alpha }\left(a_{\left(2\right)\left(1\right)}\right)a_{\left(1\right)}=a_{\left(1\right)\left(1\right)}\otimes S_{\alpha }\left(a_{\left(2\right)}\right)a_{\left(1\right)\left(2\right)}=a\otimes 1$, for all $\alpha \in \pi ,a\in H_{\alpha }$.

 (3)

$S_{{\alpha }^{-1}}{S}_{\alpha }=i{d}_{H_{\alpha }}$, for all $\alpha \in \pi$.

Proof. 

Let $\alpha \in \pi$ and $a\in H_{\alpha }.$ We have

$$\begin{array}{cc}\hfill & S_{{\alpha }^{-1}}{S}_{\alpha }\left(a\right)=S_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{\left(1\right)}{\epsilon }_{\alpha }\left(a_{\left(2\right)}\right)\right)=S_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{\left(1\right)}\right){\epsilon }_{\alpha }\left(a_{\left(2\right)}\right)=S_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{\left(1\right)}\right)\left({\epsilon }_{\alpha }\left(a_{\left(2\right)}\right)1\right)\hfill \\ \hfill =& S_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{\left(1\right)}\right)\left(S_{\alpha }\left(a_{\left(2\right)\left(1\right)}\right)a_{\left(2\right)\left(2\right)}\right)=S_1\left(a_{\left(2\right)\left(1\right)}{S}_{\alpha }\left(a_{\left(1\right)}\right)\right)a_{\left(2\right)\left(2\right)}.\hfill \end{array}$$

If $\left(1\right)$ holds, we then find that $S_{{\alpha }^{-1}}{S}_{\alpha }\left(a\right)=S_1\left(1\right)a=a$, which implies that $\left(3\right)$ holds.

If $\left(3\right)$ is satisfied, then one has

$$\begin{array}{cc}\hfill 1\otimes a=& a_{\left(1\right)}{S}_{\alpha }\left(a_{\left(2\right)\left(1\right)}\right)\otimes a_{\left(2\right)\left(2\right)}=a_{\left(1\right)}{S}_{\alpha }\left(a_{\left(2\right)\left(1\right)}\right)\otimes S_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{\left(2\right)\left(2\right)}\right)\hfill \\ \hfill =& a_{\left(1\right)}{S}_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(2\right)}\otimes S_{{\alpha }^{-1}}\left(S_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(1\right)}\right)\hfill \\ \hfill =& S_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{\left(1\right)}\right)S_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(2\right)}\otimes S_{{\alpha }^{-1}}\left(S_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(1\right)}\right)\hfill \\ \hfill =& S_{{\alpha }^{-1}}\left(S_{\alpha }{\left(a\right)}_{\left(2\right)}\right)S_{\alpha }{\left(a\right)}_{\left(1\right)\left(2\right)}\otimes S_{{\alpha }^{-1}}\left(S_{\alpha }{\left(a\right)}_{\left(1\right)\left(1\right)}\right).\hfill \end{array}$$

Applying $S_{\alpha }$ to the second tensor factor we obtain

$$\begin{array}{cc}\hfill 1\otimes S_{\alpha }\left(a\right)=& S_{{\alpha }^{-1}}\left(S_{\alpha }{\left(a\right)}_{\left(2\right)}\right)S_{\alpha }{\left(a\right)}_{\left(1\right)\left(2\right)}\otimes S_{\alpha }{S}_{{\alpha }^{-1}}\left(S_{\alpha }{\left(a\right)}_{\left(1\right)\left(1\right)}\right)\hfill \\ \hfill =& S_{{\alpha }^{-1}}\left(S_{\alpha }{\left(a\right)}_{\left(2\right)}\right)S_{\alpha }{\left(a\right)}_{\left(1\right)\left(2\right)}\otimes S_{\alpha }{\left(a\right)}_{\left(1\right)\left(1\right)}.\hfill \end{array}$$

So $\left(2\right)$ holds since S is bijective.

We have shown $\left(1\right)⟹\left(3\right)⟹\left(2\right).$

Similarly one proves $\left(2\right)⟹\left(3\right)⟹\left(1\right).$ □

Theorem 6.

Let H be a Hopf non-coassociative π-algebra with a bijective antipode S and $S^{-1}$ the composite inverse to S. Then

$$\begin{array}{ccc}\hfill S_{{\alpha }^{-1}}^{-1}\left(h_{\left(2\right)}\right)h_{\left(1\right)}=& h_{\left(2\right)}{S}_{{\alpha }^{-1}}^{-1}\left(h_{\left(1\right)}\right)={\epsilon }_{\alpha }\left(h\right)1,S_{{\left(\alpha \beta \right)}^{-1}}^{-1}\left(hg\right)=S_{{\beta }^{-1}}^{-1}\left(g\right)S_{{\alpha }^{-1}}^{-1}\left(h\right),S_1^{-1}\left(1\right)=\hfill & \hfill 1\\ \hfill {\Delta }_{{\alpha }^{-1}}\left(S_{{\alpha }^{-1}}^{-1}\left(h\right)\right)=& S_{{\alpha }^{-1}}^{-1}\left(h_{\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}^{-1}\left(h_{\left(1\right)}\right),{\epsilon }_{{\alpha }^{-1}}\left(S_{{\alpha }^{-1}}^{-1}\left(h\right)\right)={\epsilon }_{\alpha }\left(h\right)\hfill \end{array}$$

for all $\alpha ,\beta \in \pi ,$$h\in H_{\alpha }$ and $g\in H_{\beta }$.

Proof. 

The proof is straightforward. □

Theorem 7.

Let H be a Hopf non-coassociative π-algebra such that each $S_{\alpha }^{-1}$ exists, for all $\alpha \in \pi ,a\in H_{\alpha }$. Then the following identities are equivalent:

 (1)

$a_{\left(1\right)}{a}_{\left(2\right)\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)\left(2\right)}=a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\otimes a_{\left(2\right)}.$

 (2)

$a_{\left(1\right)\left(1\right)\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)}{a}_{\left(2\right)}\otimes a_{\left(1\right)\left(2\right)}=a_{\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}{a}_{\left(2\right)\left(2\right)\left(2\right)}\otimes a_{\left(2\right)\left(2\right)\left(1\right)}.$

 (3)

$a_{\left(1\right)\left(1\right)}{a}_{\left(2\right)\left(2\right)}\otimes a_{\left(1\right)\left(2\right)}\otimes a_{\left(2\right)\left(1\right)}=a_{\left(1\right)\left(1\right)}{a}_{\left(2\right)}\otimes a_{\left(1\right)\left(2\right)\left(1\right)}\otimes a_{\left(1\right)\left(2\right)\left(2\right)}.$

Proof. 

$\left(1\right)⟹\left(2\right)$ Let $T=(i{d}_{H_{\alpha }}\otimes {\sigma }_{H_{\alpha },H_{\alpha }}\otimes i{d}_{H_{\alpha }})$. Then

$$\begin{array}{cc}\hfill & a_{\left(1\right)}{a}_{\left(2\right)\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)\left(2\right)}=a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\otimes a_{\left(2\right)}\hfill \\ \hfill ⟹& (m_{\alpha ,\alpha }\otimes i{d}_{H_{\alpha }}^2)T(i{d}_{H_{\alpha }}^2\otimes {\Delta }_{\alpha })(i{d}_{H_{\alpha }}\otimes {\Delta }_{\alpha }){\Delta }_{\alpha }\left(a\right)\hfill \\ \hfill =& (m_{\alpha ,\alpha }\otimes i{d}_{H_{\alpha }}\otimes i{d}_{H_{\alpha }})T({\Delta }_{\alpha }\otimes i{d}_{H_{\alpha }}^2)({\Delta }_{\alpha }\otimes i{d}_{H_{\alpha }}){\Delta }_{\alpha }\left(a\right)\hfill \\ \hfill ⟹& (m_{\alpha ,\alpha }\otimes i{d}_{H_{\alpha }}\otimes i{d}_{H_{\alpha }})T(i{d}_{H_{\alpha }}^2\otimes {\Delta }_{\alpha })(i{d}_{H_{\alpha }}\otimes {\Delta }_{\alpha }){\Delta }_{\alpha }\left(S_{{\alpha }^{-1}}\left(b\right)\right)\hfill \\ \hfill =& (m_{\alpha ,\alpha }\otimes i{d}_{H_{\alpha }}^2)T({\Delta }_{\alpha }\otimes i{d}_{H_{\alpha }}^2)({\Delta }_{\alpha }\otimes i{d}_{H_{\alpha }}){\Delta }_{\alpha }\left(S_{{\alpha }^{-1}}\left(b\right)\right)\hfill \\ \hfill ⟹& (m_{\alpha ,\alpha }\otimes i{d}_{H_{\alpha }}^2)T(i{d}_{H_{\alpha }}^2\otimes {\Delta }_{\alpha })(i{d}_{H_{\alpha }}\otimes {\Delta }_{\alpha })(S_{{\alpha }^{-1}}\left(b_{\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)}\right))\hfill \\ \hfill =& (m_{\alpha ,\alpha }\otimes i{d}_{H_{\alpha }}\otimes i{d}_{H_{\alpha }})T({\Delta }_{\alpha }\otimes i{d}_{H_{\alpha }}^2)({\Delta }_{\alpha }\otimes i{d}_{H_{\alpha }})(S_{{\alpha }^{-1}}\left(b_{\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)}\right))\hfill \\ \hfill ⟹& (m_{\alpha ,\alpha }\otimes i{d}_{H_{\alpha }}^2)T(i{d}_{H_{\alpha }}^2\otimes {\Delta }_{\alpha })(S_{{\alpha }^{-1}}\left(b_{\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(1\right)}\right))\hfill \\ \hfill =& (m_{\alpha ,\alpha }\otimes i{d}_{H_{\alpha }}^2)T({\Delta }_{\alpha }\otimes i{d}_{H_{\alpha }}^2)(S_{{\alpha }^{-1}}\left(b_{\left(2\right)\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(2\right)\left(1\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)}\right))\hfill \\ \hfill ⟹& (m_{\alpha ,\alpha }\otimes i{d}_{H_{\alpha }}^2)T(S_{{\alpha }^{-1}}\left(b_{\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(1\right)\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(1\right)\left(1\right)}\right))\hfill \\ \hfill =& (m_{\alpha ,\alpha }\otimes i{d}_{H_{\alpha }}^2)T(S_{{\alpha }^{-1}}\left(b_{\left(2\right)\left(2\right)\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(2\right)\left(2\right)\left(1\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(2\right)\left(1\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)}\right))\hfill \\ \hfill ⟹& (m_{\alpha ,\alpha }\otimes i{d}_{H_{\alpha }}^2)(S_{{\alpha }^{-1}}\left(b_{\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(1\right)\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(1\right)\left(1\right)}\right))\hfill \\ \hfill =& (m_{\alpha ,\alpha }\otimes i{d}_{H_{\alpha }}^2)(S_{{\alpha }^{-1}}\left(b_{\left(2\right)\left(2\right)\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(2\right)\left(1\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(2\right)\left(2\right)\left(1\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)}\right))\hfill \\ \hfill ⟹& S_{{\alpha }^{-1}}\left(b_{\left(2\right)}\right)S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(1\right)\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(1\right)\left(1\right)}\right)\hfill \\ \hfill =& S_{{\alpha }^{-1}}\left(b_{\left(2\right)\left(2\right)\left(2\right)}\right)S_{{\alpha }^{-1}}\left(b_{\left(2\right)\left(1\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(2\right)\left(2\right)\left(1\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)}\right)\hfill \\ \hfill ⟹& S_{{\alpha }^{-1}{\alpha }^{-1}}\left(b_{\left(1\right)\left(1\right)\left(2\right)}{b}_{\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)\left(1\right)\left(1\right)}\right)\hfill \\ \hfill =& S_{{\alpha }^{-1}{\alpha }^{-1}}\left(b_{\left(2\right)\left(1\right)}{b}_{\left(2\right)\left(2\right)\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(2\right)\left(2\right)\left(1\right)}\right)\otimes S_{{\alpha }^{-1}}\left(b_{\left(1\right)}\right)\hfill \\ \hfill ⟹& b_{\left(1\right)\left(1\right)\left(2\right)}{b}_{\left(2\right)}\otimes b_{\left(1\right)\left(2\right)}\otimes b_{\left(1\right)\left(1\right)\left(1\right)}=b_{\left(2\right)\left(1\right)}{b}_{\left(2\right)\left(2\right)\left(2\right)}\otimes b_{\left(2\right)\left(2\right)\left(1\right)}\otimes b_{\left(1\right)}\hfill \\ \hfill ⟹& b_{\left(1\right)\left(1\right)\left(1\right)}\otimes b_{\left(1\right)\left(1\right)\left(2\right)}{b}_{\left(2\right)}\otimes b_{\left(1\right)\left(2\right)}=b_{\left(1\right)}\otimes b_{\left(2\right)\left(1\right)}{b}_{\left(2\right)\left(2\right)\left(2\right)}\otimes b_{\left(2\right)\left(2\right)\left(1\right)}\hfill \\ \hfill ⟹& a_{\left(1\right)\left(1\right)\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)}{b}_{\left(2\right)}\otimes a_{\left(1\right)\left(2\right)}=a_{\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}{b}_{\left(2\right)\left(2\right)\left(2\right)}\otimes a_{\left(2\right)\left(2\right)\left(1\right)}.\hfill \end{array}$$

Similarly, $\left(1\right)$ implies $\left(2\right).$

$\left(3\right)⟹\left(1\right)$

$$\begin{array}{cc}\hfill & a_{\left(1\right)\left(1\right)}{a}_{\left(2\right)\left(2\right)}\otimes a_{\left(1\right)\left(2\right)}\otimes a_{\left(2\right)\left(1\right)}=a_{\left(1\right)\left(1\right)}{a}_{\left(2\right)}\otimes a_{\left(1\right)\left(2\right)\left(1\right)}\otimes a_{\left(1\right)\left(2\right)\left(2\right)}\hfill \\ \hfill ⟹& a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)}{S}_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(2\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\otimes a_{\left(1\right)\left(2\right)\left(1\right)}{S}_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(1\right)}\hfill \\ \hfill =& a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}{S}_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(2\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(2\right)}{S}_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(1\right)}\hfill \\ \hfill ⟹& a_{\left(1\right)\left(1\right)\left(1\right)}{\left(a_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\right)}_{\left(2\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\otimes {\left(a_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\right)}_{\left(1\right)}\hfill \\ \hfill =& a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}{S}_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(2\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(2\right)}{S}_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(1\right)}\hfill \\ \hfill ⟹& a_{\left(1\right)}\otimes a_{\left(2\right)}\otimes 1=a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}{S}_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(2\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(2\right)}{S}_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(1\right)}\hfill \\ \hfill ⟹& a_{\left(1\right)}\otimes a_{\left(2\right)}\otimes 1=a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)\left(1\right)}\right)\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)\left(2\right)}\right)\hfill \\ \hfill ⟹& a_{\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}\otimes S_{\alpha }\left(a_{\left(2\right)\left(2\right)}\right)1=a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)\left(1\right)}\right)\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\otimes 1S_{\alpha }\left(a_{\left(2\right)\left(2\right)}\right)\hfill \\ \hfill ⟹& a_{\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}\otimes S_{\alpha }\left(a_{\left(2\right)\left(2\right)}\right)=a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)\left(1\right)}\right)\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\otimes S_{\alpha }\left(a_{\left(2\right)\left(2\right)}\right)\hfill \\ \hfill ⟹& a_{\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}\otimes S_{\alpha }^{-1}\left(S_{\alpha }\left(a_{\left(2\right)\left(2\right)}\right)\right)=a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)\left(1\right)}\right)\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\otimes S_{\alpha }^{-1}\left(S_{\alpha }\left(a_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ \hfill ⟹& a_{\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)}=a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)\left(1\right)}\right)\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\otimes a_{\left(2\right)\left(2\right)}\hfill \\ \hfill ⟹& a_{\left(1\right)}{a}_{\left(2\right)\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)\left(2\right)}=a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)\left(1\right)}\right)a_{\left(2\right)\left(2\right)\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\otimes a_{\left(2\right)\left(2\right)\left(2\right)}\hfill \\ \hfill ⟹& a_{\left(1\right)}{a}_{\left(2\right)\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)\left(2\right)}=a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}1\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\otimes a_{\left(2\right)}\hfill \\ \hfill ⟹& a_{\left(1\right)}{a}_{\left(2\right)\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)\left(2\right)}=a_{\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\otimes a_{\left(2\right)}.\hfill \end{array}$$

Similarly, $\left(1\right)$ implies $\left(3\right).$ □

We have observed that if H is a Hopf non-coassociative $\pi$-algebra with antipode S then $H^{op,cop}$ is a Hopf non-coassociative $\pi$-algebra with antipode $S^{op,cop}={\{S_{\alpha }^{op,cop}=S_{{\alpha }^{-1}}\}}_{\alpha \in \pi }$. Furthermore, the following theorem says, if $H^{op}$ or $H^{cop}$ is a Hopf non-coassociative $\pi$-algebra, then S is bijective, and vice versa.

Proposition 2.

Suppose that H is a Hopf non-coassociative π-algebra with antipode S over the field $\mathbb{k}$. Then the following are equivalent:

 (a)

$H^{op}={\{H_{\alpha }^{op}=H_{{\alpha }^{-1}}\}}_{\alpha \in \pi }$ is a Hopf non-coassociative π-algebra.

 (b)

$H^{cop}={\{H_{\alpha }^{cop}=H_{\alpha }\}}_{\alpha \in \pi }$ is a Hopf non-coassociative π-algebra.

 (c)

S is bijective.

If S is bijective, then $H^{op}$ and $H^{cop}$ have antipodes $S^{op}={\{S_{\alpha }^{op}=S_{\alpha }^{-1}\}}_{\alpha \in \pi }$ and $S^{cop}={\{S_{\alpha }^{cop}=S_{{\alpha }^{-1}}^{-1}\}}_{\alpha \in \pi }$, respectively.

Proof. 

Since $H^{cop}={\left(H^{op}\right)}^{op,cop}$ and $H^{op}={\left(H^{cop}\right)}^{op,cop}$, the parts (a) and (b) are equivalent.

If the part (c) holds, then it is easy to check that Part (a) holds. Conversely, suppose that $H^{op}$ is a Hopf non-coassociative $\pi$-algebra with antipode $T={\left\{T_{\alpha }\right\}}_{\alpha \in \pi }$. Then $T_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)}={\epsilon }_{\alpha }^{op}\left(h\right)1=h_{\left(1\right)}{T}_{\alpha }\left(h_{\left(2\right)}\right)$, or equivalently, $h_{\left(2\right)}{T}_{\alpha }\left(h_{\left(1\right)}\right)={\epsilon }_{{\alpha }^{-1}}\left(h\right)1=T_{\alpha }\left(h_{\left(2\right)}\right)h_{\left(1\right)}$, for $h\in H_{\alpha }^{op}=H_{{\alpha }^{-1}}$. Applying $S_1$ to the left-hand side of the above equation, we have

$$(S_{\alpha }\circ T_{\alpha })\left(h_{\left(1\right)}\right)S_{{\alpha }^{-1}}\left(h_{\left(2\right)}\right)={\epsilon }_{{\alpha }^{-1}}\left(h\right)1.$$

Replacing h with $S_{\alpha }\left(h\right)$ in this equation, one has

$${\epsilon }_{{\alpha }^{-1}}\left(S_{\alpha }\left(h\right)\right)1=T_{\alpha }\left(S_{\alpha }{\left(h\right)}_{\left(2\right)}\right)S_{\alpha }{\left(h\right)}_{\left(1\right)},$$

or equivalently,

$${\epsilon }_{\alpha }\left(h\right)1=T_{\alpha }\circ S_{\alpha }\left(h_{\left(1\right)}\right)S_{\alpha }\left(h_{\left(2\right)}\right).$$

Therefore $T_{\alpha }\circ S_{\alpha }$ and $S_{{\alpha }^{-1}}\circ T_{{\alpha }^{-1}}$ are both left inverses of $S_{\alpha }$ in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$. It follows from Corollary 3 that $S_{{\alpha }^{-1}}\circ T_{{\alpha }^{-1}}={\mathrm{id}}_{H_{\alpha }}=T_{\alpha }\circ S_{\alpha }$ which establishes that the part (a) implies that the part (c). □

Theorem 8.

Let H be a commutative flexible Hopf non-coassociative π-algebra. Then

$$a_{\left(1\right)}{S}_{\alpha }\left(a_{\left(2\right)\left(2\right)}\right)\otimes a_{\left(2\right)\left(1\right)}=a_{\left(1\right)\left(1\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\otimes a_{\left(1\right)\left(2\right)},\forall \alpha \in \pi ,a\in H_{\alpha }.$$

Proof. 

$\forall \alpha \in \pi ,a\in H_{\alpha }$. Since H is flexible, we have that

$$a_{\left(1\right)\left(1\right)}{a}_{\left(2\right)}\otimes a_{\left(1\right)\left(2\right)}=a_{\left(1\right)}{a}_{\left(2\right)\left(2\right)}\otimes a_{\left(2\right)\left(1\right)}$$

$$\begin{array}{cc}\hfill ⟹& \left(a_{\left(1\right)\left(1\right)}{a}_{\left(2\right)}\right)S_{\alpha }\left(a_{\left(1\right)\left(2\right)\left(2\right)}\right)\otimes a_{\left(1\right)\left(2\right)\left(1\right)}=\left(a_{\left(1\right)}{a}_{\left(2\right)\left(2\right)}\right)S_{\alpha }\left(a_{\left(2\right)\left(1\right)\left(2\right)}\right)\otimes a_{\left(2\right)\left(1\right)\left(1\right)}\hfill \\ \hfill ⟹& a_{\left(1\right)\left(1\right)}\left(a_{\left(2\right)}{S}_{\alpha }\left(a_{\left(1\right)\left(2\right)\left(2\right)}\right)\right)\otimes a_{\left(1\right)\left(2\right)\left(1\right)}=a_{\left(1\right)}\left(a_{\left(2\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)\left(1\right)\left(2\right)}\right)\right)\otimes a_{\left(2\right)\left(1\right)\left(1\right)}\hfill \\ \hfill ⟹& a_{\left(1\right)\left(1\right)}\left(S_{\alpha }\left(a_{\left(1\right)\left(2\right)\left(2\right)}\right)a_{\left(2\right)}\right)\otimes a_{\left(1\right)\left(2\right)\left(1\right)}=a_{\left(1\right)\left(1\right)\left(1\right)}\left(S_{\alpha }\left(a_{\left(1\right)\left(2\right)}\right)a_{\left(2\right)}\right)\otimes a_{\left(1\right)\left(1\right)\left(2\right)}\hfill \\ \hfill ⟹& a_{\left(1\right)\left(1\right)}{S}_{\alpha }\left(a_{\left(1\right)\left(2\right)\left(2\right)}\right)a_{\left(2\right)}\otimes a_{\left(1\right)\left(2\right)\left(1\right)}=a_{\left(1\right)}{S}_{\alpha }\left(a_{\left(2\right)\left(1\right)\left(2\right)}\right)a_{\left(2\right)\left(2\right)}\otimes a_{\left(2\right)\left(1\right)\left(1\right)}\hfill \\ \hfill ⟹& a_{\left(1\right)\left(1\right)\left(1\right)}{S}_{\alpha }\left(a_{\left(1\right)\left(1\right)\left(2\right)\left(2\right)}\right)a_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(1\right)}=a_{\left(1\right)\left(1\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\otimes a_{\left(1\right)\left(2\right)}u\hfill \end{array}$$

□

In the end of this section, we study how to construct an coassociator for any Hopf non-coassociative $\pi$-algebra.

Definition 5.

In any Hopf non-coassociative π-algebra, we define the coassociator

$$\Phi ={\{{\Phi }_{\alpha }:H_{\alpha }⟶H_{\alpha }\otimes H_{\alpha }\otimes H_{\alpha }\}}_{\alpha \in \pi }by$$

$$({\Delta }_{\alpha }\otimes i{d}_{H_{\alpha }}){\Delta }_{\alpha }\left(a\right)=(i{d}_{H_{\alpha }}\otimes {\Delta }_{\alpha }){\Delta }_{\alpha }\left(a_{\left(1\right)\left(1\right)}\right){\Phi }_{\alpha }\left(a_{\left(1\right)\left(2\right)}\right)(i{d}_{H_{{\alpha }^{-1}}}\otimes {\Delta }_{{\alpha }^{-1}}){\Delta }_{{\alpha }^{-1}}\left(S_{\alpha }\left(a_{\left(2\right)}\right)\right)$$

for all $\alpha \in \pi$ and $a\in H_{\alpha }$.

Remark 4.

For the next theorem, we will use some convenient notation. Let H be a Hopf non-coassociative π-algebra. $\forall \alpha \in \pi ,a\in H_{\alpha },$ we write ${\Phi }_{\alpha }\left(a\right)={\Phi }_a^{\left(1\right)}\otimes {\Phi }_a^{\left(2\right)}\otimes {\Phi }_a^{\left(3\right)}.$

Theorem 9.

Let H be a Hopf non-coassociative π-algebra. Then

 (1)

The associator $\Phi ={\left\{{\Phi }_{\alpha }\right\}}_{\alpha \in \pi }$ exists and is uniquely determined as

${\Phi }_{\alpha }\left(a\right)=S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(1\right)}{a}_{\left(2\right)\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(2\right)\left(2\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(2\right)\left(1\right)}{a}_{\left(2\right)\left(1\right)\left(1\right)\left(2\right)}{a}_{\left(2\right)\left(2\right)\left(2\right)\left(1\right)}$

$\otimes S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(2\right)\left(2\right)}{a}_{\left(2\right)\left(1\right)\left(2\right)}{a}_{\left(2\right)\left(2\right)\left(2\right)\left(2\right)},\forall \alpha \in \pi ,a\in H_{\alpha }$.

 (2)

$({\epsilon }_{\alpha }\otimes {\epsilon }_{\alpha }\otimes i{d}_{H_{\alpha }}){\Phi }_{\alpha }\left(a\right)=({\epsilon }_{\alpha }\otimes i{d}_{H_{\alpha }}\otimes {\epsilon }_{\alpha }){\Phi }_{\alpha }\left(a\right)$

$=(i{d}_{H_{\alpha }}\otimes {\epsilon }_{\alpha }\otimes {\epsilon }_{\alpha }){\Phi }_{\alpha }\left(a\right)=a,\forall \alpha \in \pi ,a\in H_{\alpha }.$

 (3)

${\Phi }_a^{\left(1\right)}{S}_{\alpha }\left({\Phi }_a^{\left(2\right)}\right)\otimes {\Phi }_a^{\left(3\right)}=S_{\alpha }\left({\Phi }_a^{\left(1\right)}\right){\Phi }_a^{\left(2\right)}\otimes {\Phi }_a^{\left(3\right)},\forall \alpha \in \pi ,a\in H_{\alpha }.$

 (4)

${\Phi }_a^{\left(1\right)}\otimes S_{\alpha }\left({\Phi }_a^{\left(2\right)}\right){\Phi }_a^{\left(3\right)}={\Phi }_a^{\left(1\right)}\otimes {\Phi }_a^{\left(2\right)}{S}_{\alpha }\left({\Phi }_a^{\left(3\right)}\right),\forall \alpha \in \pi ,a\in H_{\alpha }.$

 (5)

${\left({\Phi }_a^{\left(1\right)}\right)}_{\left(1\right)}{S}_{\alpha }\left({\Phi }_a^{\left(3\right)}\right)\otimes {\left({\Phi }_a^{\left(1\right)}\right)}_{\left(2\right)}{S}_{\alpha }\left({\Phi }_a^{\left(2\right)}\right)=S_{\alpha }\left({\left({\Phi }_a^{\left(1\right)}\right)}_{\left(1\right)}\right){\Phi }_a^{\left(3\right)}\otimes S_{\alpha }\left({\left({\Phi }_a^{\left(1\right)}\right)}_{\left(2\right)}\right){\Phi }_a^{\left(2\right)}$

$={\Phi }_a^{\left(1\right)}{S}_{\alpha }\left({\left({\Phi }_a^{\left(3\right)}\right)}_{\left(2\right)}\right)\otimes {\Phi }_a^{\left(2\right)}{S}_{\alpha }\left({\left({\Phi }_a^{\left(3\right)}\right)}_{\left(1\right)}\right)=S_{\alpha }\left({\Phi }_a^{\left(1\right)}\right){\left({\Phi }_a^{\left(3\right)}\right)}_{\left(2\right)}\otimes S_{\alpha }\left({\Phi }_a^{\left(2\right)}\right){\left({\Phi }_a^{\left(3\right)}\right)}_{\left(1\right)}$

$=S_{\alpha }\left({\Phi }_a^{\left(1\right)}\right){\left({\Phi }_a^{\left(2\right)}\right)}_{\left(1\right)}\otimes S_{\alpha }\left({\left({\Phi }_a^{\left(2\right)}\right)}_{\left(2\right)}\right){\Phi }_a^{\left(3\right)}=S_{\alpha }\left({\Phi }_a^{\left(1\right)}\right){\left({\Phi }_a^{\left(3\right)}\right)}_{\left(2\right)}\otimes S_{\alpha }\left({\Phi }_a^{\left(2\right)}\right){\left({\Phi }_a^{\left(3\right)}\right)}_{\left(1\right)},$$\forall \alpha \in \pi ,a\in H_{\alpha }.$

Proof. 

The proof of this theorem consists of a long tedious computation. We just show readers as follows for the part (1). The other are similar.

$\left(1\right)$$\forall \alpha \in \pi ,a\in H_{\alpha }$, we have that

$$\begin{array}{cc}\hfill & S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(1\right)}{a}_{\left(2\right)\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(2\right)\left(2\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(2\right)\left(1\right)}{a}_{\left(2\right)\left(1\right)\left(1\right)\left(2\right)}{a}_{\left(2\right)\left(2\right)\left(2\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(2\right)\left(2\right)}{a}_{\left(2\right)\left(1\right)\left(2\right)}{a}_{\left(2\right)\left(2\right)\left(2\right)\left(2\right)}\hfill \\ \hfill =& (S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(2\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(2\right)\left(2\right)})\hfill \\ \hfill & (a_{\left(2\right)\left(1\right)\left(1\right)\left(1\right)}\otimes a_{\left(2\right)\left(1\right)\left(1\right)\left(2\right)}\otimes a_{\left(2\right)\left(1\right)\left(2\right)})(a_{\left(2\right)\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)\left(2\right)\left(2\right)})\hfill \\ \hfill =& (S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(2\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(2\right)\left(2\right)})(a_{\left(2\right)\left(1\right)\left(1\right)\left(1\right)\left(1\right)}\otimes a_{\left(2\right)\left(1\right)\left(1\right)\left(1\right)\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(1\right)\left(1\right)\left(1\right)\left(2\right)\left(2\right)})\hfill \\ \hfill & {\Phi }_{\alpha }\left(a_{\left(2\right)\left(1\right)\left(1\right)\left(2\right)}\right)\hfill \\ \hfill & (S_{\alpha }{\left(a_{\left(2\right)\left(1\right)\left(2\right)}\right)}_{\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(2\right)\left(1\right)\left(2\right)}\right)}_{\left(2\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(2\right)\left(1\right)\left(2\right)}\right)}_{\left(2\right)\left(2\right)})(a_{\left(2\right)\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)\left(2\right)\left(1\right)}\otimes a_{\left(2\right)\left(2\right)\left(2\right)\left(2\right)})\hfill \\ \hfill =& (S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(1\right)}{a}_{\left(2\right)\left(1\right)\left(1\right)\left(1\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(2\right)\left(1\right)}{a}_{\left(2\right)\left(1\right)\left(1\right)\left(1\right)\left(2\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(1\right)}\right)}_{\left(2\right)\left(2\right)}{a}_{\left(2\right)\left(1\right)\left(1\right)\left(1\right)\left(2\right)\left(2\right)})\hfill \\ \hfill & {\Phi }_{\alpha }\left(a_{\left(2\right)\left(1\right)\left(1\right)\left(2\right)}\right)\hfill \\ \hfill & (S_{\alpha }{\left(a_{\left(2\right)\left(1\right)\left(2\right)}\right)}_{\left(1\right)}{a}_{\left(2\right)\left(2\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(2\right)\left(1\right)\left(2\right)}\right)}_{\left(2\right)\left(1\right)}{a}_{\left(2\right)\left(2\right)\left(2\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(2\right)\left(1\right)\left(2\right)}\right)}_{\left(2\right)\left(2\right)}{a}_{\left(2\right)\left(2\right)\left(2\right)\left(2\right)})\hfill \\ \hfill =& ({\left(S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)\left(1\right)\left(1\right)}\right)}_{\left(1\right)}\otimes {\left(S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)\left(1\right)\left(1\right)}\right)}_{\left(2\right)\left(1\right)}\otimes {\left(S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)\left(1\right)\left(1\right)}\right)}_{\left(2\right)\left(2\right)})\hfill \\ \hfill & {\Phi }_{\alpha }\left(a_{\left(2\right)\left(1\right)\left(1\right)\left(2\right)}\right)\hfill \\ \hfill & ({\left(S_{\alpha }\left(a_{\left(2\right)\left(1\right)\left(2\right)}\right)a_{\left(2\right)\left(2\right)}\right)}_{\left(1\right)}\otimes {\left(S_{\alpha }\left(a_{\left(2\right)\left(1\right)\left(2\right)}\right)a_{\left(2\right)\left(2\right)}\right)}_{\left(2\right)\left(1\right)}\otimes {\left(S_{\alpha }\left(a_{\left(2\right)\left(1\right)\left(2\right)}\right)a_{\left(2\right)\left(2\right)}\right)}_{\left(2\right)\left(2\right)})\hfill \\ \hfill =& ({\left(S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)}\right)}_{\left(1\right)}\otimes {\left(S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)}\right)}_{\left(2\right)\left(1\right)}\otimes {\left(S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)}\right)}_{\left(2\right)\left(2\right)})\hfill \\ \hfill & {\Phi }_{\alpha }\left(a_{\left(2\right)\left(2\right)}\right)(1_{\left(1\right)}\otimes 1_{\left(2\right)\left(1\right)}\otimes 1_{\left(2\right)\left(2\right)})\hfill \\ \hfill =& ({\left(S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)}\right)}_{\left(1\right)}\otimes {\left(S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)}\right)}_{\left(2\right)\left(1\right)}\otimes {\left(S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)\left(1\right)}\right)}_{\left(2\right)\left(2\right)}){\Phi }_{\alpha }\left(a_{\left(2\right)\left(2\right)}\right)\hfill \\ \hfill =& (1_{\left(1\right)}\otimes 1_{\left(2\right)\left(1\right)}\otimes 1_{\left(2\right)\left(2\right)}){\Phi }_{\alpha }\left(a\right)\hfill \\ \hfill =& {\Phi }_{\alpha }\left(a\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & (a_{\left(1\right)\left(1\right)\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(2\right)}){\Phi }_{\alpha }\left(a_{\left(1\right)\left(2\right)}\right)(S_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(2\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(2\right)\left(2\right)})\hfill \\ \hfill =& (a_{\left(1\right)\left(1\right)\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(2\right)})(S_{\alpha }{\left(a_{\left(1\right)\left(2\right)\left(1\right)}\right)}_{\left(1\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(2\right)\left(1\right)}\hfill \\ & \otimes S_{\alpha }{\left(a_{\left(1\right)\left(2\right)\left(1\right)}\right)}_{\left(2\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(1\right)\left(1\right)\left(2\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(2\right)\left(2\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(1\right)\left(2\right)\left(1\right)}\right)}_{\left(2\right)\left(2\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(1\right)\left(2\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(2\right)\left(2\right)\left(2\right)})\hfill \\ \hfill & (S_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(2\right)\left(1\right)}\otimes S_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(2\right)\left(2\right)})\hfill \\ \hfill =& a_{\left(1\right)\left(1\right)\left(1\right)}{S}_{\alpha }{\left(a_{\left(1\right)\left(2\right)\left(1\right)}\right)}_{\left(1\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(1\right)\left(1\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(2\right)\left(1\right)}{S}_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(1\right)}\hfill \\ \hfill & \otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(1\right)}{S}_{\alpha }{\left(a_{\left(1\right)\left(2\right)\left(1\right)}\right)}_{\left(2\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(1\right)\left(1\right)\left(2\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(2\right)\left(2\right)\left(1\right)}{S}_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(2\right)\left(1\right)}\hfill \\ \hfill & \otimes a_{\left(1\right)\left(1\right)\left(2\right)\left(2\right)}{S}_{\alpha }{\left(a_{\left(1\right)\left(2\right)\left(1\right)}\right)}_{\left(2\right)\left(2\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(1\right)\left(2\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(2\right)\left(2\right)\left(2\right)}{S}_{\alpha }{\left(a_{\left(2\right)}\right)}_{\left(2\right)\left(2\right)}\hfill \\ \hfill =& {\left(a_{\left(1\right)\left(1\right)}{S}_{\alpha }\left(a_{\left(1\right)\left(2\right)\left(1\right)}\right)\right)}_{\left(1\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(1\right)\left(1\right)\left(1\right)}{\left(a_{\left(1\right)\left(2\right)\left(2\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\right)}_{\left(1\right)}\hfill \\ \hfill & \otimes {\left(a_{\left(1\right)\left(1\right)}{S}_{\alpha }\left(a_{\left(1\right)\left(2\right)\left(1\right)}\right)\right)}_{\left(2\right)\left(1\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(1\right)\left(1\right)\left(2\right)}{\left(a_{\left(1\right)\left(2\right)\left(2\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\right)}_{\left(2\right)\left(1\right)}\hfill \\ \hfill & \otimes {\left(a_{\left(1\right)\left(1\right)}{S}_{\alpha }\left(a_{\left(1\right)\left(2\right)\left(1\right)}\right)\right)}_{\left(2\right)\left(2\right)}{a}_{\left(1\right)\left(2\right)\left(2\right)\left(1\right)\left(2\right)}{\left(a_{\left(1\right)\left(2\right)\left(2\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\right)}_{\left(2\right)\left(2\right)}\hfill \\ \hfill =& 1_{\left(1\right)}{a}_{\left(1\right)\left(1\right)\left(1\right)\left(1\right)}{\left(a_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\right)}_{\left(1\right)}\otimes 1_{\left(2\right)\left(1\right)}{a}_{\left(1\right)\left(1\right)\left(1\right)\left(2\right)}{\left(a_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\right)}_{\left(2\right)\left(1\right)}\hfill \\ \hfill & \otimes 1_{\left(2\right)\left(2\right)}{a}_{\left(1\right)\left(1\right)\left(2\right)}{\left(a_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\right)}_{\left(2\right)\left(2\right)}\hfill \\ \hfill =& a_{\left(1\right)\left(1\right)\left(1\right)\left(1\right)}{\left(a_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\right)}_{\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(1\right)\left(2\right)}{\left(a_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\right)}_{\left(2\right)\left(1\right)}\otimes a_{\left(1\right)\left(1\right)\left(2\right)}{\left(a_{\left(1\right)\left(2\right)}{S}_{\alpha }\left(a_{\left(2\right)}\right)\right)}_{\left(2\right)\left(2\right)}\hfill \\ \hfill =& a_{\left(1\right)\left(1\right)}{1}_{\left(1\right)}\otimes a_{\left(1\right)\left(2\right)}{1}_{\left(2\right)\left(1\right)}\otimes a_{\left(2\right)}{1}_{\left(2\right)\left(2\right)}=a_{\left(1\right)\left(1\right)}\otimes a_{\left(1\right)\left(2\right)}\otimes a_{\left(2\right)}.\hfill \end{array}$$

□

5. Crossed Hopf Non-Coassociative $\mathbf{\pi }$-Algebras

In this section we mainly study the notion of a crossed Hopf non-coassociative $\pi$-algebra and give some properties of the crossing map.

Definition 6.

A Hopf non-coassociative π-algebra $H=({\{H_{\alpha },{\Delta }_{\alpha },{\epsilon }_{\alpha }\}}_{\alpha \in \pi },m,\eta ,S)$ is said to be crossed provided it is endowed with a family $\phi ={\{{\phi }_{\beta }:H_{\alpha }⟶H_{\beta \alpha {\beta }^{-1}}\}}_{\alpha ,\beta \in \pi }$ of $\mathbb{k}$-linear maps (the cocrossing) such that

$$\begin{array}{cc}★\hfill & \hspace{1em}\hspace{1em}each{\phi }_{\beta }:H_{\alpha }⟶H_{\beta \alpha {\beta }^{-1}}is a coalgebra isomorphism,\hfill \\ ★\hfill & \hspace{1em}\hspace{1em}each{\phi }_{\beta }preserves the multiplication,i.e., for all\alpha ,\beta ,\gamma \in \pi ,\hfill \end{array}$$

(24)

$$\begin{array}{cc}★\hfill & \hspace{1em}\hspace{1em}{\phi }_{\beta }{m}_{\alpha ,\gamma }=m_{\beta \alpha {\beta }^{-1},\beta \gamma {\beta }^{-1}}({\phi }_{\beta }\otimes {\phi }_{\beta }),\hfill \end{array}$$

(25)

$$\begin{array}{cc}★\hfill & \hspace{1em}\hspace{1em}each{\phi }_{\beta }preserves the unit,i.e.,{\phi }_{\beta }\left(1\right)=1,\hfill \end{array}$$

(26)

$$\begin{array}{cc}★\hfill & \phi is multiplicative in the sense that{\phi }_{\beta {\beta }^{\prime }}={\phi }_{\beta }{\phi }_{{\beta }^{\prime }}for all\beta ,{\beta }^{\prime }\in \pi .\hfill \end{array}$$

(27)

The following result is straightforward.

Lemma 13.

Let H be a crossed Hopf non-coassociative π-algebra with cocrossing φ. Then

 (a)

${\phi }_{1|H_{\alpha }}=i{d}_{H_{\alpha }}$ for all $\alpha \in \pi ;$

 (b)

${\phi }_{\beta }^{-1}={\phi }_{{\beta }^{-1}}$ for all $\beta \in \pi ;$

 (c)

φ preserves the antipode, i.e., ${\phi }_{\beta }{S}_{\alpha }=S_{\beta \alpha {\beta }^{-1}}{\phi }_{\beta }$ for all $\alpha ,\beta \in \pi ;$

 (d)

if $\lambda ={\left({\lambda }_{\alpha }\right)}_{\alpha \in \pi }$ is a left (resp. right) π-integral in H and $\beta \in \pi$, then ${\left({\phi }_{\beta }\left({\lambda }_{{\beta }^{-1}\alpha \beta }\right)\right)}_{\alpha \in \pi }$ is also a left (resp. right) π-integral on H;

 (e)

if $g={\left(g_{\alpha }\right)}_{\alpha \in \pi }$ is a π-grouplike element of H and $\beta \in \pi$, then ${\left(g_{\beta \alpha {\beta }^{-1}}{\phi }_{\beta }\right)}_{\alpha \in \pi }$ is also a π-grouplike element of H.

Let H be a crossed Hopf non-coassociative $\pi$-algebra with cocrossing $\phi$. If the antipode of H is bijective, then the opposite (resp. coopposite) coquasigroup Hopf $\pi$-algebra to H (see Example 2) is crossed with cocrossing given by

$${\phi }_{\beta }^{op}|H_{\alpha }^{op}={\phi }_{\beta }|H_{{\alpha }^{-1}}(resp.{\phi }_{\beta }^{cop}|H_{\alpha }^{cop}={\phi }_{\beta }\left|H_{\alpha }\right)$$

for all $\alpha ,\beta \in \pi$.

Let $H=(\{H_{\alpha },m_{\alpha },1_{\alpha }\},\Delta ,\epsilon ,S,\phi )$ be a crossed Hopf non-coassociative $\pi$-algebra. Similar to ([4], Section 11.6), its mirror $\overline{H}$ is defined by the following procedure: set ${\overline{H}}_{\alpha }=H_{{\alpha }^{-1}}$ as a coalgebra, ${\overline{m}}_{\alpha ,\beta }=m_{{\beta }^{-1}{\alpha }^{-1}\beta ,{\beta }^{-1}}({\phi }_{{\beta }^{-1}}\otimes i{d}_{H_{{\beta }^{-1}}})$, $\overline{1}=1$, $\overline{S_{\alpha }}={\phi }_{\alpha }{S}_{{\alpha }^{-1}}$, ${\overline{\phi }}_{\beta }|{\overline{H}}_{\alpha }={\phi }_{\beta }|H_{{\alpha }^{-1}}$. It is also a crossed Hopf non-coassociative $\pi$-algebra.

6. Almost Cocommutative Hopf Non-Coassociative $\mathbf{\pi }$-Algebras

The aim of this section is to discuss the definition and properties of an almost cocommutative Hopf non-coassociative $\pi$-algebra and to obtain its equivalent condition.

Definition 7.

A crossed Hopf non-coassociative π-algebra $(H,\phi )$ with a bijective antipode S is called almost cocommutative if there exists a family $R={\{R_{\alpha ,\beta }\in H_{\alpha }\otimes H_{\beta }\}}_{\alpha ,\beta \in \pi }$ of invertible elements (the R-matrix) such that, for any $\alpha ,\beta ,\gamma \in \pi$ and $x\in H_{\gamma }$,

$${\Delta }_{\gamma }^{cop}\left(x\right)·({\phi }_{{\gamma }^{-1}}\otimes {\phi }_{{\gamma }^{-1}})\left(R_{\alpha ,\beta }\right)=R_{\alpha ,\beta }·{\Delta }_{\gamma }\left(x\right),$$

(28)

and the family R is invariant under the crossing, i.e., for any $\alpha ,\beta ,\gamma \in \pi$,

$$({\phi }_{\gamma }\otimes {\phi }_{\gamma })\left(R_{\alpha ,\beta }\right)=R_{\gamma \alpha {\gamma }^{-1},\gamma \beta {\gamma }^{-1}}.$$

(29)

Note that $(H_1,R_{1,1})$ is an almost cocommutative Hopf coquasigroup. It is customary to write $R_{\alpha ,\beta }^{\left(1\right)}\otimes R_{\alpha ,\beta }^{\left(2\right)}$ for $R_{\alpha ,\beta }$.

Equation (28) in Definition 7 can be written equivalently as:

$${\Delta }_{\gamma }^{cop}\left(x\right)·R_{\alpha ,\beta }=({\phi }_{\gamma }\otimes {\phi }_{\gamma })\left(R_{\alpha ,\beta }\right)·{\Delta }_{\gamma }\left(x\right),$$

for any $\alpha ,\beta ,\gamma \in \pi$ and $x\in H_{\gamma }$.

It is obvious that, for any $\alpha ,\beta ,\gamma \in \pi$,

$${\left(({\phi }_{\gamma }\otimes {\phi }_{\gamma })\left(R_{\alpha ,\beta }\right)\right)}^{-1}=({\phi }_{\gamma }\otimes {\phi }_{\gamma })\left(R_{\alpha ,\beta }^{-1}\right).$$

The family $R^{-1}$ is therefore invariant under the crossing, i.e., for any $\alpha ,\beta ,\gamma \in \pi$,

$$({\phi }_{\gamma }\otimes {\phi }_{\gamma })\left(R_{\alpha ,\beta }^{-1}\right)=R_{\gamma \alpha {\gamma }^{-1},\gamma \beta {\gamma }^{-1}}^{-1}.$$

Our first proposition generalizes the basic fact about almost cocommutative Hopf non-coassociative $\pi$-algebras.

Note that $(H_1,R_{1,1})$ is an almost cocommutative Hopf coquasigroup. It is customary to write $R_{\alpha ,\beta }^{\left(1\right)}\otimes R_{\alpha ,\beta }^{\left(2\right)}$ for $R_{\alpha ,\beta }$.

Our first proposition generalizes the basic fact about almost cocommutative Hopf non-coassociative $\pi$-algebras.

Proposition 3.

Let H be a crossed Hopf non-coassociative π-algebra, and V, W left π-modules over H, then $V\otimes W={\{V_{\alpha }\otimes W_{\alpha }\}}_{\alpha \in \pi }$ is also a left π-module over H. If H is almost cocommutative, then $V\otimes W\cong W\otimes V$ as left π-modules over H.

Proof. 

Similar as in the Hopf coquasigroup case, we define

$$h·(v\otimes w)=h_{\left(1\right)}·v\otimes h_{\left(2\right)}·w$$

for all $h\in H_{\alpha }$ and $v\in V_{\beta }$, $w\in W_{\beta }$. It is easy to see that $V\otimes W$ is a left $\pi$-module over H. If H is almost cocommutative with $R\in H\otimes H$. Then for all $v\in V_{\alpha }$, $w\in W_{\alpha }$, define

$$c_{V_{\alpha },W_{\alpha }}^{R_{1,1}}:V_{\alpha }\otimes W_{\beta }\to W_{\alpha }\otimes V_{\alpha },c_{V_{\alpha },W_{\alpha }}^{R_{1,1}}(v\otimes w)=R_{1,1}^{\left(2\right)}w\otimes R_{1,1}^{\left(1\right)}v$$

By Equation (28), $c_{V_{\alpha },W_{\alpha }}^{R_{1,1}}$ is an isomorphism with inverse given by

$${\left(c_{V_{\alpha },W_{\alpha }}^{R_{1,1}}\right)}^{-1}:W_{\alpha }\otimes V_{\alpha }\to V_{\alpha }\otimes W_{\alpha },{\left(c_{V_{\alpha },W_{\alpha }}^{R_{1,1}}\right)}^{-1}(w\otimes v)=U_{1,1}^{\left(1\right)}v\otimes U_{1,1}^{\left(2\right)}w$$

where $R_{1,1}^{-1}=U_{1,1}=U_{1,1}^{\left(1\right)}\otimes U_{1,1}^{\left(2\right)}$. □

Recall from Theorem 4 that if H be cocommutative, then $S^2=i{d}_H$. This fact can also be generalized.

Proposition 4.

Let H be an almost cocommutative Hopf non-coassociative π-algebra. Then $S^2={\{S_{{\alpha }^{-1}}\circ S_{\alpha }\}}_{\alpha \in \pi }$ is an inner automorphism of H. More precisely, let $u_{{\beta }^{-1}}=S_{\beta }\left(R_{1,\beta }^{\left(2\right)}\right)R_{1,\beta }^{\left(1\right)}$, where $R_{1,\beta }=R_{1,\beta }^{\left(1\right)}\otimes R_{1,\beta }^{\left(2\right)}$. Then, we have

 (1)

$u_{\alpha }$ is invertible, $S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)=u_{\alpha }h{u}_{\alpha }^{-1}={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)$, $S_{\alpha }\circ S_{{\alpha }^{-1}}\left(h\right)=u_{\alpha }h{u}_{\alpha }^{-1}={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)$ and $S_1\circ S_1\left(h\right)=u_{\alpha }h{u}_{\alpha }^{-1}={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)$;

 (2)

$u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)$ is relatively central for $H_{\alpha }\cup H_1\cup H_{{\alpha }^{-1}}$;

 (3)

${\phi }_{\beta }\left(u_{\alpha }\right)=u_{\beta \alpha {\beta }^{-1}}$.

Proof. 

We first show that $u_{\alpha }h=S_1\circ S_1\left(h\right)u_{\alpha }$, for all $h\in H_1$. Since H be almost cocommutative, we have

$$h_{\left(2\right)}\otimes R_{1,{\alpha }^{-1}}^{\left(1\right)}{h}_{\left(1\right)\left(1\right)}\otimes R_{1,{\alpha }^{-1}}^{\left(2\right)}{h}_{\left(1\right)\left(2\right)}=h_{\left(2\right)}\otimes h_{\left(1\right)\left(2\right)}{R}_{1,{\alpha }^{-1}}^{\left(1\right)}\otimes h_{\left(1\right)\left(1\right)}{R}_{1,{\alpha }^{-1}}^{\left(2\right)},$$

i.e.,

$$h_{\left(2\right)}\otimes R_{1,{\alpha }^{-1}}^{\left(2\right)}{h}_{\left(1\right)\left(2\right)}\otimes R_{1,{\alpha }^{-1}}^{\left(1\right)}{h}_{\left(1\right)\left(1\right)}=h_{\left(2\right)}\otimes h_{\left(1\right)\left(1\right)}{R}_{1,{\alpha }^{-1}}^{\left(2\right)}\otimes h_{\left(1\right)\left(2\right)}{R}_{1,{\alpha }^{-1}}^{\left(1\right)}.$$

Thus

$$S_1\circ S_1\left(h_{\left(2\right)}\right)S_{{\alpha }^{-1}}\left(R_{1,{\alpha }^{-1}}^{\left(2\right)}{h}_{\left(1\right)\left(2\right)}\right)R_{1,{\alpha }^{-1}}^{\left(1\right)}{h}_{\left(1\right)\left(1\right)}=S_1\circ S_1\left(h_{\left(2\right)}\right)S_{{\alpha }^{-1}}\left(h_{\left(1\right)\left(1\right)}{R}_{1,{\alpha }^{-1}}^{\left(2\right)}\right)h_{\left(1\right)\left(2\right)}{R}_{1,{\alpha }^{-1}}^{\left(1\right)}.$$

Since S is antimultiplicative, hence

$$\begin{array}{cc}& S_1\circ S_1\left(h_{\left(2\right)}\right)S_1\left(h_{\left(1\right)\left(2\right)}\right)S_{{\alpha }^{-1}}\left(R_{1,{\alpha }^{-1}}^{\left(2\right)}\right)R_{1,{\alpha }^{-1}}^{\left(1\right)}{h}_{\left(1\right)\left(1\right)}\hfill \\ \hfill =& S_1\circ S_1\left(h_{\left(2\right)}\right)S_{{\alpha }^{-1}}\left(R_{1,{\alpha }^{-1}}^{\left(2\right)}\right)S_1\left(h_{\left(1\right)\left(1\right)}\right)h_{\left(1\right)\left(2\right)}{R}_{1,{\alpha }^{-1}}^{\left(1\right)},\hfill \end{array}$$

i.e.,

$$S_1\left(h_{\left(1\right)\left(2\right)}{S}_1\left(h_{\left(2\right)}\right)\right)u_{\alpha }{h}_{\left(1\right)\left(1\right)}=S_1\circ S_1\left(h\right)u_{\alpha }.$$

Following the axiom (16) of Hopf non-coassociative $\pi$-algebra, we have

$$u_{\alpha }h=S_1\circ S_1\left(h\right)u_{\alpha },\mathrm{for}\mathrm{all}h\in H_1.$$

(30)

The following two equalities can be verified in a similar way.

$$u_{\alpha }h=S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)u_{\alpha },\mathrm{for}\mathrm{all}h\in H_{\alpha }.$$

(31)

$$u_{\alpha }h=S_{\alpha }\circ S_{{\alpha }^{-1}}\left(h\right)u_{\alpha },\mathrm{for}\mathrm{all}h\in H_{{\alpha }^{-1}}.$$

(32)

We next show that $u_{\alpha }$ is invertible. Write $R_{1,{\alpha }^{-1}}^{-1}=U_{1,\alpha }=U_{1,\alpha }^{\left(1\right)}\otimes U_{1,\alpha }^{\left(2\right)}$. Applying $m_{1,1}\circ {\sigma }_{H_1,H_1}\circ ({\mathrm{id}}_{H_1}\otimes S_1)$ to both sides of $R_{1,{\alpha }^{-1}}^{\left(1\right)}{U}_{1,\alpha }^{\left(1\right)}\otimes R_{1,{\alpha }^{-1}}^{\left(2\right)}{U}_{1,\alpha }^{\left(2\right)}=1\otimes 1$ yields $S_{\alpha }\left(U_{1,\alpha }^{\left(2\right)}\right)u_{\alpha }{U}_{1,\alpha }^{\left(1\right)}=1$ from which $S_{\alpha }\left(U_{1,\alpha }^{\left(2\right)}\right)S_1\circ S_1\left(U_{1,\alpha }^{\left(1\right)}\right)u_{\alpha }=1$ follows by Equation (30). Observe that we have not used the fact that S is bijective at this point. Since S is bijective we can use Equation (32) to calculate $1=S_{\alpha }\left(U_{1,\alpha }^{\left(2\right)}\right)u_{\alpha }{U}_{1,\alpha }^{\left(1\right)}=S_{\alpha }\circ S_{{\alpha }^{-1}}\circ S_{{\alpha }^{-1}}^{-1}\left(U_{1,\alpha }^{\left(2\right)}\right)u_{\alpha }{U}_{1,\alpha }^{\left(1\right)}=u_{\alpha }{S}_{{\alpha }^{-1}}^{-1}\left(U_{1,\alpha }^{\left(2\right)}\right)U_{1,\alpha }^{\left(1\right)}$. We have shown that $u_{\alpha }$ has a left inverse and a right inverse. $u_{\alpha }$ is therefore invertible. By Equations (30)–(32), the three equations below can be therefore deduced:

$$\begin{array}{cc}\hfill S_1\circ S_1\left(h\right)& =u_{\alpha }h{u}_{\alpha }^{-1},\mathrm{for}\mathrm{all}h\in H_1.\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)& =u_{\alpha }h{u}_{\alpha }^{-1},\mathrm{for}\mathrm{all}h\in H_{\alpha }.\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill S_{\alpha }\circ S_{{\alpha }^{-1}}\left(h\right)& =u_{\alpha }h{u}_{\alpha }^{-1},\mathrm{for}\mathrm{all}h\in H_{{\alpha }^{-1}}.\hfill \end{array}$$

(35)

Applying $S_1$ to Equation (33) and replacing h by $S_1^{-1}\left(h\right)$ yields the formula $S_1\circ S_1\left(h\right)={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)$.

Applying $S_{\alpha }$ to Equation (34) and replacing h by $S_{\alpha }^{-1}\left(h\right)$ gives rise to the formula $S_{\alpha }\circ S_{{\alpha }^{-1}}\left(h\right)={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)$.

Applying $S_{{\alpha }^{-1}}$ to Equation (35) and replacing h by $S_{{\alpha }^{-1}}^{-1}\left(h\right)$ gives birth to the formula $S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)$.

To check that $u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)$ is relatively central for $H_{\alpha }$, we will prove that for all $g\in H_{\alpha }$, $g{u}_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)g$.

Let $h=S_{\alpha }\left(u_{\alpha }\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)$, then

$$S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)=u_{\alpha }h{u}_{\alpha }^{-1}=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)u_{\alpha }^{-1}$$

and

$$S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}{S}_{\alpha }\left(u_{\alpha }\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)S_{\alpha }\left(u_{\alpha }\right)=g.$$

So

$$g=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)u_{\alpha }^{-1}$$

i.e., $g{u}_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)g$ for all $g\in H_{\alpha }$.

To check that $u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)$ is relatively central for $H_1$, we will prove that for all $g\in H_1$, $g{u}_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)g$.

Let $h=S_{\alpha }\left(u_{\alpha }\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)$, then

$$S_1\circ S_1\left(h\right)=u_{\alpha }h{u}_{\alpha }^{-1}=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)u_{\alpha }^{-1}$$

and

$$S_1\circ S_1\left(h\right)={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}{S}_{\alpha }\left(u_{\alpha }\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)S_{\alpha }\left(u_{\alpha }\right)=g.$$

So

$$g=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)u_{\alpha }^{-1}$$

i.e., $g{u}_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)g$ for all $g\in H_1$.

To check that $u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)$ is relatively central for $H_{{\alpha }^{-1}}$, we will prove that for all $g\in H_{{\alpha }^{-1}}$, $g{u}_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)g$.

Let $h=S_{\alpha }\left(u_{\alpha }\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)$, then

$$S_{\alpha }\circ S_{{\alpha }^{-1}}\left(h\right)=u_{\alpha }h{u}_{\alpha }^{-1}=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)u_{\alpha }^{-1}$$

and

$$S_{\alpha }\circ S_{{\alpha }^{-1}}\left(h\right)={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}{S}_{\alpha }\left(u_{\alpha }\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)S_{\alpha }\left(u_{\alpha }\right)=g.$$

So

$$g=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)u_{\alpha }^{-1}$$

i.e., $g{u}_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)g$ for all $g\in H_{{\alpha }^{-1}}$.

$$\begin{array}{cc}\hfill {\phi }_{\beta }\left(u_{\alpha }\right)& ={\phi }_{\beta }\left(S_{{\alpha }^{-1}}\left(R_{1,{\alpha }^{-1}}^{\left(2\right)}\right)R_{1,{\alpha }^{-1}}^{\left(1\right)}\right)={\phi }_{\beta }\left(S_{{\alpha }^{-1}}\left(R_{1,{\alpha }^{-1}}^{\left(2\right)}\right)\right){\phi }_{\beta }\left(R_{1,{\alpha }^{-1}}^{\left(1\right)}\right)\hfill \\ \hfill & =S_{\beta {\alpha }^{-1}{\beta }^{-1}}\left({\phi }_{\beta }\left(R_{1,{\alpha }^{-1}}^{\left(2\right)}\right)\right){\phi }_{\beta }\left(R_{1,{\alpha }^{-1}}^{\left(1\right)}\right)=S_{\beta {\alpha }^{-1}{\beta }^{-1}}\left(R_{1,\beta {\alpha }^{-1}{\beta }^{-1}}^{\left(2\right)}\right)R_{1,\beta {\alpha }^{-1}{\beta }^{-1}}^{\left(1\right)}=u_{\beta \alpha {\beta }^{-1}}.\hfill \end{array}$$

This completes the proof. □

Corollary 8.

 (1)

$S_{{\alpha }^{-1}}\circ S_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }$;

 (2)

$S_{\alpha }\circ S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)=u_{\alpha }^{-1}$;

 (3)

$S_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)u_{\alpha }^{-1}$. In particular, $u_{\alpha }$ and $S_{\alpha }\left(u_{\alpha }\right)$ commute;

 (4)

$S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)=u_{\alpha }{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)u_{\alpha }^{-1}$. In particular, $u_{\alpha }$ and $S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)$ commute;

 (5)

${\phi }_{\beta }\left(u_{\alpha }^{-1}\right)=u_{\beta \alpha {\beta }^{-1}}^{-1}$.

Proof. 

Part (1) is straightforward from Proposition 4. Apropos of part (2), we calculate as follows:

$$S_{{\alpha }^{-1}}\circ S_{\alpha }\left(u_{\alpha }\right)S_{\alpha }\circ S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)=S_1\circ S_1\left(u_{\alpha }{u}_{\alpha }^{-1}\right)=S_1\circ S_1\left(1\right)=1,$$

and also

$$S_{\alpha }\circ S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)S_{{\alpha }^{-1}}\circ S_{\alpha }\left(u_{\alpha }\right)=S_1\circ S_1\left(u_{\alpha }^{-1}{u}_{\alpha }\right)=S_1\circ S_1\left(1\right)=1.$$

Thus $S_{{\alpha }^{-1}}\circ S_{\alpha }\left(u_{\alpha }\right)$ and $S_{\alpha }\circ S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)$ are inverses, from which $S_{\alpha }\circ S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)=u_{\alpha }^{-1}$.

To show part (3), we use part (1) and Proposition 4 to calculate

$$S_{\alpha }\left(u_{\alpha }\right)=S_{\alpha }\circ S_{{\alpha }^{-1}}\circ S_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)u_{\alpha }^{-1},$$

whereby $u_{\alpha }$ and $S_{\alpha }\left(u_{\alpha }\right)$ commute.

To establish part (4), we use part (3) to make the following calculation:

$$S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}={\left(u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)u_{\alpha }^{-1}\right)}^{-1}=u_{\alpha }{\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}{u}_{\alpha }^{-1}=u_{\alpha }{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)u_{\alpha }^{-1},$$

whereby $u_{\alpha }$ and $S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)$ commute.

It remains to check part (5). Observe that

$${\phi }_{\beta }\left(u_{\alpha }^{-1}\right){\phi }_{\beta }\left(u_{\alpha }\right)={\phi }_{\beta }\left(u_{\alpha }^{-1}{u}_{\alpha }\right)={\phi }_{\beta }\left(1\right)=1$$

and also that

$${\phi }_{\beta }\left(u_{\alpha }\right){\phi }_{\beta }\left(u_{\alpha }^{-1}\right)={\phi }_{\beta }\left(u_{\alpha }{u}_{\alpha }^{-1}\right)={\phi }_{\beta }\left(1\right)=1.$$

Thus ${\phi }_{\beta }\left(u_{\alpha }\right)$ and ${\phi }_{\beta }\left(u_{\alpha }^{-1}\right)$ are inverses.

It follows from Proposition 4 that

$${\phi }_{\beta }\left(u_{\alpha }^{-1}\right)={\left({\phi }_{\beta }\left(u_{\alpha }\right)\right)}^{-1}=u_{\beta \alpha {\beta }^{-1}}^{-1}.$$

□

Corollary 9.

$S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)=u_{\alpha }h{\phi }_{{\beta }^{-1}}\left(u_{\alpha }^{-1}\right)={\phi }_{\beta }\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)h{S}_{\alpha }\left(u_{\alpha }\right)$.

Proof. 

We first show that $u_{\alpha }h=S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right){\phi }_{{\beta }^{-1}}\left(u_{\alpha }\right)$, for all $h\in H_{\beta }$. Since H is almost cocommutative, we have

$$h_{\left(2\right)}\otimes R_{1,{\alpha }^{-1}}^{\left(1\right)}{h}_{\left(1\right)\left(1\right)}\otimes R_{1,{\alpha }^{-1}}^{\left(2\right)}{h}_{\left(1\right)\left(2\right)}=h_{\left(2\right)}\otimes h_{\left(1\right)\left(2\right)}{\phi }_{{\beta }^{-1}}\left(R_{1,{\alpha }^{-1}}^{\left(1\right)}\right)\otimes h_{\left(1\right)\left(1\right)}{\phi }_{{\beta }^{-1}}\left(R_{1,{\alpha }^{-1}}^{\left(2\right)}\right),$$

i.e.,

$$h_{\left(2\right)}\otimes R_{1,{\alpha }^{-1}}^{\left(2\right)}{h}_{\left(1\right)\left(2\right)}\otimes R_{1,{\alpha }^{-1}}^{\left(1\right)}{h}_{\left(1\right)\left(1\right)}=h_{\left(2\right)}\otimes h_{\left(1\right)\left(1\right)}{R}_{1,{\beta }^{-1}{\alpha }^{-1}\beta }^{\left(2\right)}\otimes h_{\left(1\right)\left(2\right)}{R}_{1,{\beta }^{-1}{\alpha }^{-1}\beta }^{\left(1\right)}.$$

Thus

$$\begin{array}{cc}& \phantom{=}{S}_{{\beta }^{-1}}\circ S_{\beta }\left(h_{\left(2\right)}\right)S_{{\alpha }^{-1}\beta }\left(R_{1,{\alpha }^{-1}}^{\left(2\right)}{h}_{\left(1\right)\left(2\right)}\right)R_{1,{\alpha }^{-1}}^{\left(1\right)}{h}_{\left(1\right)\left(1\right)}\hfill \\ & =S_{{\beta }^{-1}}\circ S_{\beta }\left(h_{\left(2\right)}\right)S_{{\alpha }^{-1}\beta }\left(h_{\left(1\right)\left(1\right)}{R}_{1,{\beta }^{-1}{\alpha }^{-1}\beta }^{\left(2\right)}\right)h_{\left(1\right)\left(2\right)}{R}_{1,{\beta }^{-1}{\alpha }^{-1}\beta }^{\left(1\right)}.\hfill \end{array}$$

Using that S is antimultiplicative we have

$$\begin{array}{cc}& S_{{\beta }^{-1}}\circ S_{\beta }\left(h_{\left(2\right)}\right)S_{\beta }\left(h_{\left(1\right)\left(2\right)}\right)S_{{\alpha }^{-1}}\left(R_{1,{\alpha }^{-1}}^{\left(2\right)}\right)R_{1,{\alpha }^{-1}}^{\left(1\right)}{h}_{\left(1\right)\left(1\right)}\hfill \\ \hfill =& S_{{\beta }^{-1}}\circ S_{\beta }\left(h_{\left(2\right)}\right)S_{{\beta }^{-1}{\alpha }^{-1}\beta }\left(R_{1,{\beta }^{-1}{\alpha }^{-1}\beta }^{\left(2\right)}\right)S_{\beta }\left(h_{\left(1\right)\left(1\right)}\right)h_{\left(1\right)\left(2\right)}{R}_{1,{\beta }^{-1}{\alpha }^{-1}\beta }^{\left(1\right)}\hfill \end{array}$$

i.e.,

$$S_1\left(h_{\left(1\right)\left(2\right)}{S}_{\beta }\left(h_{\left(2\right)}\right)\right)u_{\alpha }{h}_{\left(1\right)\left(1\right)}=S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)u_{{\beta }^{-1}\alpha \beta }.$$

Following the axiom (16) of coquasigroup Hopf $\pi$-algebra, we have

$$u_{\alpha }h=S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)u_{{\beta }^{-1}\alpha \beta },\mathrm{for}\mathrm{all}h\in H_{\beta }.$$

It follows that

$$S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)=u_{\alpha }h{u}_{{\beta }^{-1}\alpha \beta }^{-1}=u_{\alpha }h{\left({\phi }_{{\beta }^{-1}}\left(u_{\alpha }\right)\right)}^{-1}=u_{\alpha }h{\phi }_{{\beta }^{-1}}\left(u_{\alpha }^{-1}\right),\mathrm{for}\mathrm{all}h\in H_{\beta }.$$

Applying $S_{\beta }$ to this expression and replacing h by $S_{\beta }^{-1}\left(h\right)$ yields the following calculation:

$$\begin{array}{cc}\hfill S_{\beta }\circ & S_{{\beta }^{-1}}\left(h\right)={\left(S_{{\beta }^{-1}\alpha \beta }\left(u_{{\beta }^{-1}\alpha \beta }\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)={\left(S_{{\beta }^{-1}\alpha \beta }\left({\phi }_{{\beta }^{-1}}\left(u_{\alpha }\right)\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)\hfill \\ \hfill =& {\left({\phi }_{{\beta }^{-1}}\left(S_{\alpha }\left(u_{\alpha }\right)\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)={\phi }_{{\beta }^{-1}}\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)h{S}_{\alpha }\left(u_{\alpha }\right),\hfill \end{array}$$

or equivalently, $S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)={\phi }_{\beta }\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)h{S}_{\alpha }\left(u_{\alpha }\right)$. □

Corollary 10.

For any $\alpha \in \pi$, $g{\phi }_{{\beta }^{-1}}\left(u_{\alpha }\right)S_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }{\phi }_{\beta }\left(S_{\alpha }\left(u_{\alpha }\right)\right)g$ for all $g\in H_{\beta }$. In particular, $u_1{S}_1\left(u_1\right)$ is a central element of H.

Proof. 

Let $h={\phi }_{\beta }\left(S_{\alpha }\left(u_{\alpha }\right)\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)$, then

$$S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)=u_{\alpha }h{\phi }_{{\beta }^{-1}}\left(u_{\alpha }^{-1}\right)=u_{\alpha }{\phi }_{\beta }\left(S_{\alpha }\left(u_{\alpha }\right)\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right){\phi }_{{\beta }^{-1}}\left(u_{\alpha }^{-1}\right)$$

and

$$\begin{array}{cc}\hfill S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)=& {\phi }_{\beta }\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)h{S}_{\alpha }\left(u_{\alpha }\right)\hfill \\ \hfill =& {\phi }_{\beta }\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right){\phi }_{\beta }\left(S_{\alpha }\left(u_{\alpha }\right)\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)S_{\alpha }\left(u_{\alpha }\right)\hfill \\ \hfill =& {\phi }_{\beta }\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}{S}_{\alpha }\left(u_{\alpha }\right)\right)g{S}_1\left(u_{\alpha }{u}_{\alpha }^{-1}\right)={\phi }_{\beta }\left(1\right)g{S}_1\left(1\right)=g.\hfill \end{array}$$

So

$$g=u_{\alpha }{\phi }_{\beta }\left(S_{\alpha }\left(u_{\alpha }\right)\right)g{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right){\phi }_{{\beta }^{-1}}\left(u_{\alpha }^{-1}\right)$$

i.e., $g{\phi }_{{\beta }^{-1}}\left(u_{\alpha }\right)S_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }{\phi }_{\beta }\left(S_{\alpha }\left(u_{\alpha }\right)\right)g$ for all $g\in H_{\beta }$. □

It is well-known that the two equivalent conditions for a Hopf coquasigroup to be almost cocommutative have been obtained in [17]. Next in a similar way we will prove one equivalent condition for a Hopf non-coassociative $\pi$-algebra to be almost cocommutative.

Set ${\overline{H}}_{\alpha }=H_{{\alpha }^{-1}}$, ${\overline{m}}_{\alpha ,\beta }=m_{{\alpha }^{-1},{\beta }^{-1}}^{op}=m_{{\beta }^{-1},{\alpha }^{-1}}\circ {\sigma }_{H_{{\alpha }^{-1}},H_{{\beta }^{-1}}}$, ${\overline{\Delta }}_{\alpha }={\Delta }_{{\alpha }^{-1}}^{cop}$, ${\overline{\epsilon }}_{\alpha }={\epsilon }_{{\alpha }^{-1}}$ and ${\overline{S}}_{\alpha }=S_{{\alpha }^{-1}}$. Recall from the statement (3) in Example 2 that

$$\overline{H}=\left({\left\{{\overline{H}}_{\alpha }\right\}}_{\alpha \in \pi },\overline{m}={\left\{{\overline{m}}_{\alpha ,\beta }\right\}}_{\alpha ,\beta \in \pi },1,\overline{\Delta }={\left\{{\overline{\Delta }}_{\alpha }\right\}}_{\alpha \in \pi },{\overline{\epsilon }}_{\alpha },\overline{S}={\left\{{\overline{S}}_{\alpha }\right\}}_{\alpha \in \pi }\right)$$

is again a Hopf non-coassociative $\pi$-algebra where we write ${\overline{m}}_{\alpha ,\beta }(a\otimes b)=a·b=ba$.

We can now define $\pi$-module actions of $\overline{H}={\left\{{\overline{H}}_{\alpha }\right\}}_{\alpha \in \pi }={\left\{H_{{\alpha }^{-1}}\right\}}_{\alpha \in \pi }$ on $H^{\ast }$ by

$$(h\rightharpoonup p)\left(g\right)=p(g·h)\mathrm{and}(q\leftharpoonup h)\left(g\right)=q(h·g)$$

for all $g\in {\overline{H}}_{{\alpha }^{-1}}=H_{\alpha }$, $h\in {\overline{H}}_{{\beta }^{-1}}=H_{\beta }$ and $p\in H_{\beta \alpha }^{\ast }$, $q\in H_{\alpha \beta }^{\ast }$.

Fix $\gamma \in \pi$, and define $\pi$-module actions of H on ${\left\{{Hom}_{\mathbb{k}}\left(H_{\alpha }^{\ast },H_{\alpha \gamma }\right)\right\}}_{\alpha \in \pi }$ by

$$\left(h\stackrel{\gamma }{\to }f\right)\left(p\right)=f\left(h_{\left(1\right)}\rightharpoonup p\right)·h_{\left(2\right)}\mathrm{and}\left(f\stackrel{\gamma }{\leftarrow }h\right)\left(q\right)={\phi }_{{\gamma }^{-1}}\left(h_{\left(1\right)}·{\phi }_{\gamma }\circ f\left(q\leftharpoonup h_{\left(2\right)}\right)\right)$$

for all $h\in H_{\beta }$, $p\in H_{\beta \alpha }^{\ast }$, $q\in H_{\alpha \beta }^{\ast }$ and $f\in {Hom}_{\mathbb{k}}\left(H_{\alpha }^{\ast },H_{\alpha \gamma }\right)$.

It is easy to check that

$$h\rightharpoonup (p\leftharpoonup g)=(h\rightharpoonup p)\leftharpoonup g$$

whereby

$$h\stackrel{\gamma }{\to }\left(f\stackrel{\gamma }{\leftarrow }g\right)=\left(h\stackrel{\gamma }{\to }f\right)\stackrel{\gamma }{\leftarrow }g$$

for all $p\in H_{\alpha }^{\ast }$ and $f\in {Hom}_{\mathbb{k}}\left(H_{\alpha }^{\ast },H_{\alpha \gamma }\right)$.

Therefore, we can define

$$h\gamma f=h_{\left(1\right)}\stackrel{\gamma }{\to }f\stackrel{\gamma }{\leftarrow }{S}_{\beta }\left(h_{\left(2\right)}\right)\in {Hom}_{\mathbb{k}}\left(H_{\beta \alpha {\beta }^{-1}}^{\ast },H_{\beta \alpha {\beta }^{-1}\gamma }\right)$$

for all $h\in H_{\beta }$ and $f\in {Hom}_{\mathbb{k}}\left(H_{\alpha }^{\ast },H_{\alpha \gamma }\right)$. It is obvious that

$$g\gamma \left(h\gamma f\right)=\left(gh\right)\gamma fand1\gamma f=f.$$

Next we will prove that there is a close relationship between the $\pi$-module actions $\stackrel{\gamma }{\to }$ and $\stackrel{\gamma }{\leftarrow }$ of H on ${\left\{{Hom}_{\mathbb{k}}\left(H_{\alpha }^{\ast },H_{\alpha \gamma }\right)\right\}}_{\alpha \in \pi }$.

Lemma 14.

We have $h\stackrel{\gamma }{\to }f=\left(h_{\left(1\right)}\gamma f\right)\stackrel{\gamma }{\leftarrow }{h}_{\left(2\right)}$, for all $h\in H_{\beta }$ and $f\in {Hom}_{\mathbb{k}}\left(H_{\alpha }^{\ast },H_{\alpha \gamma }\right)$.

Proof. 

Let $h\in H_{\beta }$ and $f\in {Hom}_{\mathbb{k}}\left(H_{\alpha }^{\ast },H_{\alpha \gamma }\right)$, then

$$\begin{array}{cc}\hfill \left(h_{\left(1\right)}\gamma f\right)\stackrel{\gamma }{\leftarrow }{h}_{\left(2\right)}& =\left(h_{\left(1\right)\left(1\right)}\stackrel{\gamma }{\to }f\stackrel{\gamma }{\leftarrow }{S}_{\beta }\left(h_{\left(1\right)\left(2\right)}\right)\right)\stackrel{\gamma }{\leftarrow }{h}_{\left(2\right)}\hfill \\ & =\left(h_{\left(1\right)\left(1\right)}\stackrel{\gamma }{\to }f\right)\stackrel{\gamma }{\leftarrow }\left(S_{\beta }\left(h_{\left(1\right)\left(2\right)}\right)h_{\left(2\right)}\right)=h\stackrel{\gamma }{\to }f\stackrel{\gamma }{\leftarrow }1=h\stackrel{\gamma }{\to }f.\hfill \end{array}$$

The third equality follows from the axioms of a coquasigroup Hopf $\pi$-algebra. □

Now, we give an equivalent condition for a Hopf non-coassociative $\pi$-algebra to be almost cocommutative, provided the family R is invariant under the crossing.

Proposition 5.

Let H be a Hopf non-coassociative π-algebra and $R={\{R_{\alpha ,\beta }=R_{\alpha ,\beta }^{\left(1\right)}\otimes R_{\alpha ,\beta }^{\left(2\right)}\in H_{\alpha }\otimes H_{\beta }\}}_{\alpha ,\beta \in \pi }$. Define $f\in {Hom}_{\mathbb{k}}\left(H_{\alpha }^{\ast },H_{\alpha \gamma }\right)$ by $f\left(p\right)=p\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right){\phi }_{{\gamma }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)$, $\forall p\in H_{\alpha }^{\ast }$. Give ${\left\{{Hom}_{\mathbb{k}}\left(H_{\alpha }^{\ast },H_{\beta }\right)\right\}}_{\alpha \in \pi }$ the π-module structures over H described above. Then the following are equivalent:

 (1)

for all $\alpha ,\beta ,\gamma \in \pi$ and $h\in H_{\gamma }$, we have

$$h_{\left(2\right)}{\phi }_{{\gamma }^{-1}}\left(R_{\alpha ,\beta }^{\left(1\right)}\right)\otimes h_{\left(1\right)}{\phi }_{{\gamma }^{-1}}\left(R_{\alpha ,\beta }^{\left(2\right)}\right)=R_{\alpha ,\beta }^{\left(1\right)}{h}_{\left(1\right)}\otimes R_{\alpha ,\beta }^{\left(2\right)}{h}_{\left(2\right)};$$

 (2)

for all $h\in H_{\beta }$ and $f\in {Hom}_{\mathbb{k}}\left(H_{\alpha }^{\ast },H_{\alpha \gamma }\right)$, we have

$$\left(f\stackrel{\gamma }{\leftarrow }h\right)={\phi }_{{\gamma }^{-1}}\circ \left(h\underset{\gamma }{\to }\tilde{f}\right),$$

where $\underset{\gamma }{\to }$ is formally similar to the π-module action $\stackrel{\gamma }{\to }$ and $\tilde{f}$ is an associated function defined by $\tilde{f}\left(p\right)=p\left({\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right)\right){\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)$, for any $p\in H_{{\beta }^{-1}\alpha \beta }^{\ast }$.

Proof. 

$\left(1\right)\Rightarrow \left(2\right)$ For all $h\in H_{\beta }$, $p\in H_{\alpha \beta }^{\ast }$ and $f\in {Hom}_{\mathbb{k}}\left(H_{\alpha }^{\ast },H_{\alpha \gamma }\right)$,

$$\begin{array}{cc}\hfill \left(f\stackrel{\gamma }{\leftarrow }h\right)\left(p\right)& ={\phi }_{{\gamma }^{-1}}\left(h_{\left(1\right)}·{\phi }_{\gamma }\circ f\left(p\leftharpoonup h_{\left(2\right)}\right)\right)\hfill \\ & ={\phi }_{{\gamma }^{-1}}\left(h_{\left(1\right)}·{\phi }_{\gamma }\left(\left(p\leftharpoonup h_{\left(2\right)}\right)\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right){\phi }_{{\gamma }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)\right)\right)\hfill \\ & ={\phi }_{{\gamma }^{-1}}\left(h_{\left(1\right)}·{\phi }_{\gamma }\left(p\left(h_{\left(2\right)}·R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right){\phi }_{{\gamma }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)\right)\right)\hfill \\ & =p\left(h_{\left(2\right)}·R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right){\phi }_{{\gamma }^{-1}}\left(h_{\left(1\right)}·{\phi }_{\gamma }\left({\phi }_{{\gamma }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)\right)\right)\hfill \\ & =p\left(h_{\left(2\right)}·R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right){\phi }_{{\gamma }^{-1}}\left(h_{\left(1\right)}·R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)\hfill \\ & =p\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}{h}_{\left(2\right)}\right){\phi }_{{\gamma }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}{h}_{\left(1\right)}\right).\hfill \end{array}$$

Since $h_{\left(2\right)}{\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)\otimes h_{\left(1\right)}{\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right)=R_{\gamma \alpha ,\alpha }^{\left(1\right)}{h}_{\left(1\right)}\otimes R_{\gamma \alpha ,\alpha }^{\left(2\right)}{h}_{\left(2\right)}$, thus

$$\begin{array}{cc}& p\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}{h}_{\left(2\right)}\right){\phi }_{{\gamma }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}{h}_{\left(1\right)}\right)\hfill \\ \hfill =& p\left(h_{\left(1\right)}{\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right)\right){\phi }_{{\gamma }^{-1}}\left(h_{\left(2\right)}{\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)\right).\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill \left(f\stackrel{\gamma }{\leftarrow }h\right)\left(p\right)& =p\left(h_{\left(1\right)}{\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right)\right){\phi }_{{\gamma }^{-1}}\left(h_{\left(2\right)}{\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)\right)\hfill \\ & =p\left({\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right)·h_{\left(1\right)}\right){\phi }_{{\gamma }^{-1}}\left({\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)·h_{\left(2\right)}\right)\hfill \\ & ={\phi }_{{\gamma }^{-1}}\left(p\left({\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right)·h_{\left(1\right)}\right){\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)·h_{\left(2\right)}\right)\hfill \\ & ={\phi }_{{\gamma }^{-1}}\left(\left(h_{\left(1\right)}\rightharpoonup p\right)\left({\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right)\right){\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)·h_{\left(2\right)}\right)\hfill \\ & ={\phi }_{{\gamma }^{-1}}\left(\tilde{f}\left(h_{\left(1\right)}\rightharpoonup p\right)·h_{\left(2\right)}\right)\hfill \\ & ={\phi }_{{\gamma }^{-1}}\circ \left(h\underset{\gamma }{\to }\tilde{f}\right)\left(p\right),\hfill \end{array}$$

for all $h\in H_{\beta }$, $p\in H_{\alpha \beta }^{\ast }$ and $\tilde{f}\in {Hom}_{\mathbb{k}}\left(H_{{\beta }^{-1}\alpha \beta }^{\ast },H_{{\beta }^{-1}\gamma \alpha \beta }\right)$.

$\left(2\right)\Rightarrow \left(1\right)$ For all $h\in H_{\beta }$, $p\in H_{\alpha \beta }^{\ast }$ and $f\in {Hom}_{\mathbb{k}}\left(H_{\alpha }^{\ast },H_{\alpha \gamma }\right)$, we have

$$\left(f\stackrel{\gamma }{\leftarrow }h\right)\left(p\right)={\phi }_{{\gamma }^{-1}}\circ \left(h\underset{\gamma }{\to }\tilde{f}\right)\left(p\right).$$

Thus

$$p\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}{h}_{\left(2\right)}\right){\phi }_{{\gamma }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}{h}_{\left(1\right)}\right)=p\left(h_{\left(1\right)}{\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right)\right){\phi }_{{\gamma }^{-1}}\left(h_{\left(2\right)}{\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)\right),$$

i.e.,

$$(p\otimes {\phi }_{{\gamma }^{-1}})\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}{h}_{\left(2\right)}\otimes R_{\gamma \alpha ,\alpha }^{\left(1\right)}{h}_{\left(1\right)}\right)=(p\otimes {\phi }_{{\gamma }^{-1}})\left(h_{\left(1\right)}{\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right)\otimes h_{\left(2\right)}{\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)\right).$$

Then we have

$$R_{\gamma \alpha ,\alpha }^{\left(2\right)}{h}_{\left(2\right)}\otimes R_{\gamma \alpha ,\alpha }^{\left(1\right)}{h}_{\left(1\right)}=h_{\left(1\right)}{\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(2\right)}\right)\otimes h_{\left(2\right)}{\phi }_{{\beta }^{-1}}\left(R_{\gamma \alpha ,\alpha }^{\left(1\right)}\right)$$

This completes the proof. □

The following corollary is a direct conclusion.

Corollary 11.

Let H be an almost cocommutative Hopf non-coassociative π-algebra with an invertible antipode S. Then $h1f={\epsilon }_1\left(h\right)f$, for all $h\in H_1$.

7. Quasitriangular Hopf Non-Coassociative $\mathbf{\pi }$-Algebras

In the current section, we will introduce and discuss the definition of a quasitriangular Hopf non-coassociative $\pi$-algebra and study its main properties. We construct a new Turaev’s braided monoidal category $Re{p}_{\pi }\left(H\right)$ over a quasitriangular Hopf non-coassociative $\pi$-algebra H.

Definition 8.

A quasitriangular Hopf non-coassociative π-algebra is a crossed Hopf non-coassociative π-algebra $(H,\phi )$ with a family $\mathcal{R}={\{{\mathcal{R}}_{\alpha ,\beta }\in H_{\alpha }\otimes H_{\beta }\}}_{\alpha ,\beta \in \pi }$ of elements (the R-matrix) satisfying Equations (28) and (29) such that, for any $\alpha ,\beta ,\gamma \in \pi$,

$$\begin{array}{cc}\hfill (i{d}_{H_{\alpha \beta }}\otimes {\Delta }_{\gamma })\left({\mathcal{R}}_{\alpha \beta ,\gamma }\right)& ={\left({\mathcal{R}}_{\alpha ,\gamma }\right)}_{13}·{\left({\mathcal{R}}_{\beta ,\gamma }\right)}_{12},\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill ({\Delta }_{\alpha }\otimes i{d}_{H_{\beta \gamma }})\left({\mathcal{R}}_{\alpha ,\beta \gamma }\right)& ={\left({\mathcal{R}}_{\alpha ,\beta }\right)}_{13}·{\left({\mathcal{R}}_{\alpha ,\gamma }\right)}_{23},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill ({\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_1}){\mathcal{R}}_{\alpha ,1}& =1,\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill ({\mathrm{id}}_{H_1}\otimes {\epsilon }_{\alpha }){\mathcal{R}}_{1,\alpha }& =1\hfill \end{array}$$

(39)

where, for $\mathbb{k}$-spaces P, Q and $r={\Sigma }_j{p}_j\otimes q_j\in P\otimes Q$, we set $r_{12}=r\otimes 1\in P\otimes Q\otimes H_1,$ $r_{23}=1\otimes r\in H_1\otimes P\otimes Q$ and $r_{13}={\Sigma }_j{p}_j\otimes 1\otimes q_j\in P\otimes H_1\otimes Q$.

Note that ${\mathcal{R}}_{1,1}$ is a (classical) R-matrix for the Hopf coquasigroup $H_1$.

We find that a quasitriangular Hopf non-coassociative $\pi$-algebra also constructs a solution to the generalized quantum Yang–Baxter equation and a much stronger property of its antipode holds which are similar as a quasitriangular Hopf coquasigroup in [23].

Example 3.

Let H be a quasitriangular Hopf non-coassociative π-algebra with R-matrix $R={\left\{R_{\alpha ,\beta }\right\}}_{\alpha ,\beta \in \pi }$.

 (1)

We can consider the coopposite crossed Hopf non-coassociative π-algebra $H^{op}$ to H. It is quasitriangular by setting ${\mathcal{R}}_{\alpha ,\beta }^{op}=(S_{\alpha }\otimes i{d}_{H_{{\beta }^{-1}}})\left({\mathcal{R}}_{\alpha ,{\beta }^{-1}}\right)$.

 (2)

Consider again the coopposite crossed Hopf non-coassociative π-algebra $H^{op}$ to H. It is quasitriangular by setting ${\mathcal{R}}_{\alpha ,\beta }^{op}={\sigma }_{{\beta }^{-1},{\alpha }^{-1}}\left({\mathcal{R}}_{{\beta }^{-1},{\alpha }^{-1}}\right)$.

Lemma 15.

If $(H,\mathcal{R})$ is quasitriangular, then the following additional properties hold:

 (1)

$\left(1\otimes ({\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }}){\mathcal{R}}_{\alpha ,\beta }\right)·{\mathcal{R}}_{\alpha ,\gamma }={\mathcal{R}}_{\alpha ,\beta \gamma }$;

 (2)

${\mathcal{R}}_{\alpha ,\beta }·\left(1\otimes ({\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_{\gamma }}){\mathcal{R}}_{\alpha ,\gamma }\right)={\mathcal{R}}_{\alpha ,\beta \gamma }$;

 (3)

$\left(({\mathrm{id}}_{H_{\alpha }}\otimes {\epsilon }_{\gamma }){\mathcal{R}}_{\alpha ,\gamma }\otimes 1\right)·{\mathcal{R}}_{\beta ,\gamma }={\mathcal{R}}_{\alpha \beta ,\gamma }$;

 (4)

${\mathcal{R}}_{\alpha ,\gamma }·\left(({\mathrm{id}}_{H_{\beta }}\otimes {\epsilon }_{\gamma }){\mathcal{R}}_{\beta ,\gamma }\otimes 1\right)={\mathcal{R}}_{\alpha \beta ,\gamma }$.

Proof. 

We only need to show part (1) since the proof of other parts is similar. Applying ${\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_{\alpha }}\otimes {\mathrm{id}}_{H_{\beta \gamma }}$ to both sides of Equation (37), we obtain ${\mathcal{R}}_{\alpha ,\beta \gamma }=({\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_{\alpha }}\otimes {\mathrm{id}}_{H_{\beta \gamma }})({\Delta }_{\alpha }\otimes {\mathrm{id}}_{H_{\beta \gamma }})\left({\mathcal{R}}_{\alpha ,\beta \gamma }\right)=({\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_{\alpha }}\otimes {\mathrm{id}}_{H_{\beta \gamma }})\left({\left({\mathcal{R}}_{\alpha ,\beta }\right)}_{13}·{\left({\mathcal{R}}_{\alpha ,\gamma }\right)}_{23}\right)$ whence $\left(1\otimes ({\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }}){\mathcal{R}}_{\alpha ,\beta }\right)·{\mathcal{R}}_{\alpha ,\gamma }={\mathcal{R}}_{\alpha ,\beta \gamma }$. □

Lemma 16.

Let $(H,\mathcal{R})$ be a quasitriangular Hopf non-coassociative π-algebra, and write ${\mathcal{R}}_{\alpha ,\beta }={\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}\otimes {\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}$. Then, for any $\alpha \in \pi$, ${\mathcal{R}}_{\alpha ,1}$ is invertible. More precisely, we have ${\mathcal{R}}_{\alpha ,1}^{-1}=(S_{\alpha }\otimes {\mathrm{id}}_{H_1}){\mathcal{R}}_{\alpha ,1}$.

Proof. 

Using Equation (38) and applying $(m_{{\alpha }^{-1},\alpha }\otimes {\mathrm{id}}_{H_1})\circ (S_{\alpha }\otimes {\mathrm{id}}_{H_{\alpha }}\otimes {\mathrm{id}}_{H_1})$ and $(m_{\alpha ,{\alpha }^{-1}}\otimes {\mathrm{id}}_{H_1})\circ ({\mathrm{id}}_{H_{\alpha }}\otimes S_{\alpha }\otimes {\mathrm{id}}_{H_1})$ to both sides of Equation (37), we obtain

$$1\otimes 1=1{\epsilon }_{\alpha }\left({\mathcal{R}}_{\alpha ,1}^{\left(1\right)}\right)\otimes {\mathcal{R}}_{\alpha ,1}^{\left(2\right)}=S_{\alpha }\left({\mathcal{R}}_{\alpha ,1}^{\left(1\right)}\right){\widehat{\mathcal{R}}}_{\alpha ,1}^{\left(1\right)}\otimes {\mathcal{R}}_{\alpha ,1}^{\left(2\right)}{\widehat{\mathcal{R}}}_{\alpha ,1}^{\left(2\right)}$$

and

$$1\otimes 1=1{\epsilon }_{\alpha }\left({\mathcal{R}}_{\alpha ,1}^{\left(1\right)}\right)\otimes {\mathcal{R}}_{\alpha ,1}^{\left(2\right)}={\mathcal{R}}_{\alpha ,1}^{\left(1\right)}{S}_{\alpha }\left({\widehat{\mathcal{R}}}_{\alpha ,1}^{\left(1\right)}\right)\otimes {\mathcal{R}}_{\alpha ,1}^{\left(2\right)}{\widehat{\mathcal{R}}}_{\alpha ,1}^{\left(2\right)}$$

where ${\widehat{\mathcal{R}}}_{\alpha ,1}={\mathcal{R}}_{\alpha ,1}$. Thus ${\mathcal{R}}_{\alpha ,1}$ and $(S_{\alpha }\otimes {\mathrm{id}}_{H_1}){\mathcal{R}}_{\alpha ,1}$ are inverses. □

Theorem 10.

Let $(H,\mathcal{R})$ be a quasitriangular Hopf non-coassociative π-algebra, and write ${\mathcal{R}}_{\alpha ,\beta }={\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}\otimes {\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}$. Then $\mathcal{R}$ is invertible and ${\mathcal{R}}_{\alpha ,{\beta }^{-1}}^{-1}=(S_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }}){\mathcal{R}}_{\alpha ,\beta }$.

Proof. 

Applying $(m_{{\alpha }^{-1},\alpha }\otimes {\mathrm{id}}_{H_{\beta }})\circ (S_{\alpha }\otimes {\mathrm{id}}_{H_{\alpha }}\otimes {\mathrm{id}}_{H_{\beta }})$ to both sides of Equation (37) yields:

$$1\otimes {\epsilon }_{\alpha }\left({\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}\right){\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}=S_{\alpha }\left({\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}\right){\mathcal{R}}_{\alpha ,1}^{\left(1\right)}\otimes {\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}{\mathcal{R}}_{\alpha ,1}^{\left(2\right)},$$

or equivalently,

$$1\otimes ({\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }}){\mathcal{R}}_{\alpha ,\beta }=(S_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }})\left({\mathcal{R}}_{\alpha ,\beta }\right)·{\mathcal{R}}_{\alpha ,1}.$$

Multiplying both sides on the left by ${\mathcal{R}}_{\alpha ,{\beta }^{-1}}$, by using Lemma 15, we obtain

$${\mathcal{R}}_{\alpha ,1}={\mathcal{R}}_{\alpha ,{\beta }^{-1}}·(S_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }})\left({\mathcal{R}}_{\alpha ,\beta }\right)·{\mathcal{R}}_{\alpha ,1}.$$

Hence, ${\mathcal{R}}_{\alpha ,{\beta }^{-1}}·(S_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }})\left({\mathcal{R}}_{\alpha ,\beta }\right)=1$ follows by the invertiblity of ${\mathcal{R}}_{\alpha ,1}$.

Applying $(m_{\alpha ,{\alpha }^{-1}}\otimes {\mathrm{id}}_{H_{\beta }})\circ ({\mathrm{id}}_{H_{\alpha }}\otimes S_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }})$ to both sides of Equation (37) yields:

$$1\otimes {\epsilon }_{\alpha }\left({\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}\right){\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}={\mathcal{R}}_{\alpha ,1}^{\left(1\right)}{S}_{\alpha }\left({\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}\right)\otimes {\mathcal{R}}_{\alpha ,1}^{\left(2\right)}{\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)},$$

or equivalently,

$$1\otimes ({\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }}){\mathcal{R}}_{\alpha ,\beta }={\mathcal{R}}_{\alpha ,1}·(S_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }})\left({\mathcal{R}}_{\alpha ,\beta }\right).$$

Multiplying both sides on the right by ${\mathcal{R}}_{\alpha ,{\beta }^{-1}}$, by using Lemma 15, we obtain

$${\mathcal{R}}_{\alpha ,1}={\mathcal{R}}_{\alpha ,1}·(S_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }})\left({\mathcal{R}}_{\alpha ,\beta }\right)·{\mathcal{R}}_{\alpha ,{\beta }^{-1}}.$$

Hence, $(S_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }})\left({\mathcal{R}}_{\alpha ,\beta }\right)·{\mathcal{R}}_{\alpha ,{\beta }^{-1}}=1$ follows by the invertiblity of ${\mathcal{R}}_{\alpha ,1}$.

Therefore ${\mathcal{R}}_{\alpha ,{\beta }^{-1}}$ is invertible and ${\mathcal{R}}_{\alpha ,{\beta }^{-1}}^{-1}=(S_{\alpha }\otimes {\mathrm{id}}_{H_{\beta }})\left({\mathcal{R}}_{\alpha ,\beta }\right)$. □

Theorem 11.

If $(H,\mathcal{R})$ is quasitriangular, then the following additional properties hold:

 (1)

$(S_{\beta }\otimes S_{\gamma }){\mathcal{R}}_{\beta ,\gamma }={\mathcal{R}}_{{\beta }^{-1},{\gamma }^{-1}}$;

 (2)

${\mathcal{R}}_{1,\alpha }=\left({\mathrm{id}}_{H_1}\otimes S_{{\alpha }^{-1}}\right){\mathcal{R}}_{1,\alpha }^{-1}$;

 (3)

${\mathcal{R}}_{{\alpha }^{-1},\beta }=\left({\mathrm{id}}_{H_{{\alpha }^{-1}}}\otimes S_{{\beta }^{-1}}\right){\mathcal{R}}_{\alpha ,\beta }^{-1}$;

 (4)

$\mathcal{R}$ satisfies the generalized quantum Yang–Baxter equation:

$${\left({\mathcal{R}}_{\delta ,\lambda }\right)}_{12}{\left({\mathcal{R}}_{\alpha ,\beta }\right)}_{13}{\left({\mathcal{R}}_{\alpha ,\gamma }\right)}_{23}={\left({\mathcal{R}}_{\alpha ,\beta }\right)}_{23}{\left({\mathcal{R}}_{\alpha ,\gamma }\right)}_{13}{\left(({\phi }_{{\alpha }^{-1}}\otimes {\phi }_{{\alpha }^{-1}}){\mathcal{R}}_{\delta ,\lambda }\right)}_{12}.$$

Proof. 

We first establish part (1). Using Lemma 15 and Theorem 10, we apply $(S_{\alpha \beta }\otimes {\mathrm{id}}_{H_1})\circ ({\mathrm{id}}_{H_{\alpha \beta }}\otimes m_{{\gamma }^{-1},\gamma })\circ ({\mathrm{id}}_{H_{\alpha \beta }}\otimes S_{\gamma }\otimes {\mathrm{id}}_{H_{\gamma }})$ to both sides of Equation (36) to obtain

$$\begin{array}{cc}& (S_{\alpha \beta }\otimes S_1)\left({\mathcal{R}}_{\alpha ,\gamma }·{\mathcal{R}}_{{\beta }^{-1},\gamma }^{-1}\right)=(S_{\alpha \beta }\otimes S_1)\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(1\right)}{\epsilon }_{\gamma }\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}\right)\otimes 1\right)\hfill \\ \hfill =& S_{\alpha \beta }\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(1\right)}{\epsilon }_{\gamma }\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}\right)\right)\otimes 1=S_{\alpha \beta }\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(1\right)}\right)\otimes S_{\gamma }\left({\mathcal{R}}_{\alpha \beta ,\gamma \left(1\right)}^{\left(2\right)}\right){\mathcal{R}}_{\alpha \beta ,\gamma \left(2\right)}^{\left(2\right)}\hfill \\ \hfill =& S_{\beta }\left({\mathcal{R}}_{\beta ,\gamma }^{\left(1\right)}\right)S_{\alpha }\left({\mathcal{R}}_{\alpha ,\gamma }^{\left(1\right)}\right)\otimes S_{\gamma }\left({\mathcal{R}}_{\beta ,\gamma }^{\left(2\right)}\right){\mathcal{R}}_{\alpha ,\gamma }^{\left(2\right)}=(S_{\beta }\otimes S_{\gamma }){\mathcal{R}}_{\beta ,\gamma }·\left(S_{\alpha }\otimes {\mathrm{id}}_{H_{\gamma }}\right){\mathcal{R}}_{\alpha ,\gamma }\hfill \\ \hfill =& (S_{\beta }\otimes S_{\gamma }){\mathcal{R}}_{\beta ,\gamma }·{\mathcal{R}}_{\alpha ,{\gamma }^{-1}}^{-1},\hfill \end{array}$$

i.e., $(S_{\beta }\otimes S_{\gamma }){\mathcal{R}}_{\beta ,\gamma }=(S_{\alpha \beta }\otimes S_1)\left({\mathcal{R}}_{\alpha ,\gamma }·{\mathcal{R}}_{{\beta }^{-1},\gamma }^{-1}\right)·{\mathcal{R}}_{\alpha ,{\gamma }^{-1}}$. Thus part (1) follows by setting $\alpha ={\beta }^{-1}$.

Parts (2) and (3) follow directly from part (1) and Theorem 10.

To show part (4), we use Equation (37) to calculate

$$\begin{array}{cc}\hfill & {\left({\mathcal{R}}_{\delta ,\lambda }\right)}_{12}{\left({\mathcal{R}}_{\alpha ,\beta }\right)}_{13}{\left({\mathcal{R}}_{\alpha ,\gamma }\right)}_{23}={\left({\mathcal{R}}_{\delta ,\lambda }\right)}_{12}({\Delta }_{\alpha }\otimes i{d}_{H_{\beta \gamma }})\left({\mathcal{R}}_{\alpha ,\beta \gamma }\right)\hfill \\ \hfill =& {\mathcal{R}}_{\delta ,\lambda }{\Delta }_{\alpha }\left({\mathcal{R}}_{\alpha ,\beta \gamma }^{\left(1\right)}\right)\otimes {\mathcal{R}}_{\alpha ,\beta \gamma }^{\left(2\right)}={\Delta }_{\alpha }^{cop}\left({\mathcal{R}}_{\alpha ,\beta \gamma }^{\left(1\right)}\right)({\phi }_{{\alpha }^{-1}}\otimes {\phi }_{{\alpha }^{-1}}){\mathcal{R}}_{\delta ,\lambda }\otimes {\mathcal{R}}_{\alpha ,\beta \gamma }^{\left(2\right)}\hfill \\ \hfill =& {\left({\mathcal{R}}_{\alpha ,\beta }\right)}_{23}{\left({\mathcal{R}}_{\alpha ,\gamma }\right)}_{13}{\left(({\phi }_{{\alpha }^{-1}}\otimes {\phi }_{{\alpha }^{-1}}){\mathcal{R}}_{\delta ,\lambda }\right)}_{12}.\hfill \end{array}$$

Thus $\mathcal{R}$ satisfies the generalized quantum Yang–Baxter equation. □

Proposition 6.

Let $(H,\mathcal{R})$ be a quasitriangular Hopf non-coassociative π-algebra. For any $\alpha \in \pi$, set $u_{\alpha }=S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}$. Then $u_{\alpha }$ is invertible, $u_{\alpha }^{-1}={\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}{S}_1\circ S_1\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}\right)$, $S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)=u_{\alpha }h{u}_{{\beta }^{-1}\alpha \beta }^{-1}=u_{\alpha }h{\left({\phi }_{{\beta }^{-1}}\left(u_{\alpha }\right)\right)}^{-1}=u_{\alpha }h\left({\phi }_{{\beta }^{-1}}\left(u_{\alpha }^{-1}\right)\right)$, ${\phi }_{\beta }\left(u_{\alpha }\right)=u_{\beta \alpha {\beta }^{-1}}$ and ${\phi }_{\beta }\left(u_{\alpha }^{-1}\right)=u_{\beta \alpha {\beta }^{-1}}^{-1}$, for all $h\in H_{\beta }$.

Proof. 

The calculations in the proof of Corollary 9 and Proposition 4 showed that $u_{\alpha }h=S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)u_{{\beta }^{-1}\alpha \beta }$ holds and $S_{\alpha }\left(U_{1,\alpha }^{\left(2\right)}\right)S_1\circ S_1\left(U_{1,\alpha }^{\left(1\right)}\right)u_{\alpha }=1$ where ${\mathcal{R}}_{1,{\alpha }^{-1}}^{-1}=U_{1,\alpha }^{\left(1\right)}\otimes U_{1,\alpha }^{\left(2\right)}$ as well as ${\phi }_{\beta }\left(u_{\alpha }\right)=u_{\beta \alpha {\beta }^{-1}}$. Let $v_{{\alpha }^{-1}}=S_{\alpha }\left(U_{1,\alpha }^{\left(2\right)}\right)S_1\circ S_1\left(U_{1,\alpha }^{\left(1\right)}\right)$. Then $v_{{\alpha }^{-1}}{u}_{\alpha }=1$ and $v_{{\alpha }^{-1}}=S_{\alpha }\left(U_{1,\alpha }^{\left(2\right)}\right)S_1\circ S_1\left(U_{1,\alpha }^{\left(1\right)}\right)=S_{\alpha }\left({\mathcal{R}}_{1,\alpha }^{\left(2\right)}\right)S_1\circ S_1\circ S_1\left({\mathcal{R}}_{1,\alpha }^{\left(1\right)}\right)={\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}{S}_1\circ S_1\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}\right)$ by Theorems 10 and 11. Let $h_{\alpha }=S_1\left(U_{1,\alpha }^{\left(1\right)}\right)U_{1,\alpha }^{\left(2\right)}$. Then $v_{{\alpha }^{-1}}=S_{\alpha }\left(h_{\alpha }\right)$. Now $v_{{\alpha }^{-1}}=S_{\alpha }\circ S_{{\alpha }^{-1}}\left(v_{{\alpha }^{-1}}\right)$ by Theorem 11. Therefore, $v_{{\alpha }^{-1}}=S_{\alpha }\circ S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h_{\alpha }\right)$. Since

$$\begin{array}{cc}\hfill S_{\alpha }\circ S_{{\alpha }^{-1}}\circ & S_{\alpha }\left(h_{\alpha }\right)u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }{S}_{\alpha }\left(h_{\alpha }\right)S_{\alpha }\left(u_{\alpha }\right)=u_{\alpha }{S}_{{\alpha }^2}\left(u_{\alpha }{h}_{\alpha }\right)\hfill \\ & =u_{\alpha }{S}_{{\alpha }^2}(S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h_{\alpha }\right)u_{\alpha })=u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)S_{\alpha }\circ S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h_{\alpha }\right),\hfill \end{array}$$

it follows that $v_{{\alpha }^{-1}}$ and $u_{\alpha }S\left(u_{\alpha }\right)$ commute. Consequently

$$\begin{array}{cc}\hfill u_{\alpha }& \left(S_{\alpha }\left(u_{\alpha }\right)v_{{\alpha }^{-1}}{S}_{{\alpha }^{-1}}\left(v_{{\alpha }^{-1}}\right)\right)=\left(\left(u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)\right)v_{{\alpha }^{-1}}\right)S_{{\alpha }^{-1}}\left(v_{{\alpha }^{-1}}\right)\hfill \\ & =\left(v_{{\alpha }^{-1}}\left(u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)\right)\right)S_{{\alpha }^{-1}}\left(v_{{\alpha }^{-1}}\right)=\left(v_{{\alpha }^{-1}}{u}_{\alpha }\right)\left(S_{\alpha }\left(u_{\alpha }\right)S_{{\alpha }^{-1}}\left(v_{{\alpha }^{-1}}\right)\right)\hfill \\ & =\left(v_{{\alpha }^{-1}}{u}_{\alpha }\right)S_1\left(v_{{\alpha }^{-1}}{u}_{\alpha }\right)=1S_1\left(1\right)=1.\hfill \end{array}$$

We have shown that $u_{\alpha }$ has a left inverse $v_{{\alpha }^{-1}}$ and also has a right inverse. Therefore $u_{\alpha }$ is invertible. As $u_{\alpha }h=S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)u_{{\beta }^{-1}\alpha \beta }$ and ${\phi }_{\beta }\left(u_{\alpha }\right)=u_{\beta \alpha {\beta }^{-1}}$ hold, our proof is complete. □

Definition 9.

Let $(H,\mathcal{R})$ be a quasitriangular Hopf non-coassociative π-algebra over $\mathbb{k}$. The Drinfel’d element of $(H,\mathcal{R})$ is the element $u={\left\{u_{\alpha }\right\}}_{\alpha \in \pi }$ of Proposition 6. The quantum Casimir element of H is the family ${\left\{u_{\alpha }{S}_{\alpha }\left(u_{\alpha }\right)\right\}}_{\alpha \in \pi }$ of products.

Theorem 12.

Let $(H,\mathcal{R})$ be a quasitriangular Hopf non-coassociative π-algebra with the antipode S. Then S is bijective; thus H is almost cocommutative.

Proof. 

We set $T_{\alpha }\left(h\right)=u_{\alpha }^{-1}{S}_{{\alpha }^{-1}}\left(h\right)u_{\alpha }$. Using Proposition 6 we have

$$\begin{array}{cc}\hfill T_{\alpha }\left(h_{\left(1\right)}\right)·h_{\left(2\right)\left(1\right)}& \otimes h_{\left(2\right)\left(2\right)}=h_{\left(2\right)\left(1\right)}{T}_{\alpha }\left(h_{\left(1\right)}\right)\otimes h_{\left(2\right)\left(2\right)}=h_{\left(2\right)\left(1\right)}{u}_{\alpha }^{-1}{S}_{{\alpha }^{-1}}\left(h_{\left(1\right)}\right)u_{\alpha }\otimes h_{\left(2\right)\left(2\right)}\hfill \\ \hfill =& u_{\alpha }^{-1}{S}_{\alpha }\circ S_{{\alpha }^{-1}}\left(h_{\left(2\right)\left(1\right)}\right)S_{{\alpha }^{-1}}\left(h_{\left(1\right)}\right)u_{\alpha }\otimes h_{\left(2\right)\left(2\right)}\hfill \\ \hfill =& u_{\alpha }^{-1}{S}_1\left(h_{\left(1\right)}{S}_{{\alpha }^{-1}}\left(h_{\left(2\right)\left(1\right)}\right)\right)u_{\alpha }\otimes h_{\left(2\right)\left(2\right)}=u_{\alpha }^{-1}{S}_1\left(1\right)u_{\alpha }\otimes h=u_{\alpha }^{-1}1u_{\alpha }\otimes h=1\otimes h,\hfill \end{array}$$

and similarly for $h_{\left(1\right)}·T_{\alpha }\left(h_{\left(2\right)\left(1\right)}\right)\otimes h_{\left(2\right)\left(2\right)}=1\otimes h$, $h_{\left(1\right)\left(1\right)}\otimes T_{\alpha }\left(h_{\left(1\right)\left(2\right)}\right)·h_{\left(2\right)}=h\otimes 1$, and $h_{\left(1\right)\left(1\right)}\otimes h_{\left(1\right)\left(2\right)}·T_{\alpha }\left(h_{\left(2\right)}\right)=h\otimes 1$.

This means that $T={\left\{T_{\alpha }\right\}}_{\alpha \in \pi }$ is an antipode on $H^{op}$ and hence the inverse of the antipode S on H according to Proposition 2. □

The following reconciles the original definition of quasitriangular Hopf non-coassociative $\pi$-algebra with the one given here.

Proposition 7.

Let H be a crossed Hopf non-coassociative π-algebra over $\mathbb{k}$ and $\mathcal{R}={\left\{{\mathcal{R}}_{\alpha ,\beta }\right\}}_{\alpha ,\beta \in \pi }$. Then the following are equivalent:

 (a)

$(H,\mathcal{R})$ is quasitriangular.

 (b)

H is almost cocommutative, where $\mathcal{R}$ is invertible and satisfies Equations (36) and (37).

Proof. 

Part (a) implies part (b) by definition and Theorem 12. Suppose that the hypothesis of part (b) holds. We only need to show that Equations (38) and (39) hold. Applying ${\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_{\alpha }}\otimes {\mathrm{id}}_{H_1}$ to both sides of Equation (37), we obtain ${\mathcal{R}}_{\alpha ,1}=({\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_{\alpha }}\otimes {\mathrm{id}}_{H_1})({\Delta }_{\alpha }\otimes {\mathrm{id}}_{H_1})\left({\mathcal{R}}_{\alpha ,1}\right)=({\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_{\alpha }}\otimes {\mathrm{id}}_{H_1})\left({\left({\mathcal{R}}_{\alpha ,1}\right)}_{13}·{\left({\mathcal{R}}_{\alpha ,1}\right)}_{23}\right)$ whence $({\epsilon }_{\alpha }\otimes {\mathrm{id}}_{H_1}){\mathcal{R}}_{\alpha ,1}=1$ since ${\mathcal{R}}_{\alpha ,1}$ is invertible. Similarly for $({\mathrm{id}}_{H_1}\otimes {\epsilon }_{\alpha }){\mathcal{R}}_{1,\alpha }=1$. □

What the entire preceding discussion illustrates is the following equivalent characterization for a quasitriangular Hopf non-coassociative $\pi$-algebra:

Definition 10.

A quasitriangular Hopf non-coassociative π-algebra is a crossed Hopf non-coassociative π-algebra $(H,\phi )$ with a family $\mathcal{R}={\{{\mathcal{R}}_{\alpha ,\beta }\in H_{\alpha }\otimes H_{\beta }\}}_{\alpha ,\beta \in \pi }$ of invertible elements (the R-matrix) satisfying Equations (28), (29), (36) and (37).

Corollary 12.

Let $(H,\mathcal{R})$ be a quasitriangular Hopf non-coassociative π-algebra with a bijective antipode S, then

 (1)

$u_{\alpha }^{-1}=S_{{\alpha }^{-1}}^{-1}{S}_{\alpha }^{-1}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}$;

 (2)

${(S_{{\alpha }^{-1}}\circ S_{\alpha })}^2\left(h\right)=g_{\alpha }h{g}_{\alpha }^{-1}$ for all $h\in H_{\alpha }$, where $u_{\alpha }=S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}$ and $g_{\alpha }=u_{\alpha }{\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}$;

 (3)

${(S_{\alpha }\circ S_{{\alpha }^{-1}})}^2\left(h\right)=g_{\alpha }h{g}_{\alpha }^{-1}$ for all $h\in H_{{\alpha }^{-1}}$, where $u_{\alpha }=S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}$ and $g_{\alpha }=u_{\alpha }{\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}$;

 (4)

$S_1^4\left(h\right)=g_{\alpha }h{g}_{\alpha }^{-1}$ for all $h\in H_1$, where $u_{\alpha }=S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}$ and $g_{\alpha }=u_{\alpha }{\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}$;

 (5)

${\epsilon }_{{\alpha }^2}\left(g_{\alpha }\right)=1$ and ${\phi }_{\beta }\left(g_{\alpha }\right)=g_{\beta \alpha {\beta }^{-1}}$.

Proof. 

Apropos of part (1). Write ${\mathcal{R}}_{1,{\alpha }^{-1}}^{-1}=U_{1,\alpha }^{\left(1\right)}\otimes U_{1,\alpha }^{\left(2\right)}$. Consider the calculation:

$$\begin{array}{cc}\hfill u_{\alpha }{S}_{{\alpha }^{-1}}^{-1}\left(U_{1,\alpha }^{\left(2\right)}\right)U_{1,\alpha }^{\left(1\right)}& =S_{\alpha }\circ S_{{\alpha }^{-1}}\left(S_{{\alpha }^{-1}}^{-1}\left(U_{1,\alpha }^{\left(2\right)}\right)\right)u_{\alpha }{U}_{1,\alpha }^{\left(1\right)}\hfill \\ & =S_{\alpha }\circ S_{{\alpha }^{-1}}\left(S_{{\alpha }^{-1}}^{-1}\left(U_{1,\alpha }^{\left(2\right)}\right)\right)S_{{\alpha }^{-1}}\left(R_{1,{\alpha }^{-1}}^{\left(2\right)}\right)R_{1,{\alpha }^{-1}}^{\left(1\right)}{U}_{1,\alpha }^{\left(1\right)}\hfill \\ & =S_{\alpha }\left(U_{1,\alpha }^{\left(2\right)}\right)S_{{\alpha }^{-1}}\left(R_{1,{\alpha }^{-1}}^{\left(2\right)}\right)R_{1,{\alpha }^{-1}}^{\left(1\right)}{U}_{1,\alpha }^{\left(1\right)}\hfill \\ & =S_1\left(R_{1,{\alpha }^{-1}}^{\left(2\right)}{U}_{1,\alpha }^{\left(2\right)}\right)R_{1,{\alpha }^{-1}}^{\left(1\right)}{U}_{1,\alpha }^{\left(1\right)}=1,\hfill \end{array}$$

from which we obtain $u_{\alpha }^{-1}=S_{{\alpha }^{-1}}^{-1}\left(U_{1,\alpha }^{\left(2\right)}\right)U_{1,\alpha }^{\left(1\right)}$. We use Theorem 11 to obtain ${\mathcal{R}}_{1,{\alpha }^{-1}}=\left({\mathrm{id}}_{H_1}\otimes S_{\alpha }\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{-1}=U_{1,\alpha }^{\left(1\right)}\otimes S_{\alpha }\left(U_{1,\alpha }^{\left(2\right)}\right)$, or equivalently $\left({\mathrm{id}}_{H_1}\otimes S_{\alpha }^{-1}\right){\mathcal{R}}_{1,{\alpha }^{-1}}=U_{1,\alpha }^{\left(1\right)}\otimes U_{1,\alpha }^{\left(2\right)}$ by the bijectivity of S, thus leading to the formula:

$$u_{\alpha }^{-1}=S_{{\alpha }^{-1}}^{-1}{S}_{\alpha }^{-1}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}.$$

To establish part (2), observe from Proposition 4 that $u_{\alpha }\in H_{\alpha }$ is invertible and $S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)=u_{\alpha }h{u}_{\alpha }^{-1}={\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)$ for all $h\in H_{\alpha }$, then

$$\begin{array}{cc}\hfill S_{{\alpha }^{-1}}\circ S_{\alpha }\circ S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)=& S_{{\alpha }^{-1}}\circ S_{\alpha }\left(u_{\alpha }h{u}_{\alpha }^{-1}\right)\hfill \\ \hfill =& S_{{\alpha }^{-1}}\circ S_{\alpha }\left(u_{\alpha }\right)S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)S_{\alpha }\circ S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)\hfill \\ \hfill =& u_{\alpha }{\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}h{S}_{\alpha }\left(u_{\alpha }\right)u_{\alpha }^{-1},\hfill \end{array}$$

or equivalently $S_{{\alpha }^{-1}}\circ S_{\alpha }\circ S_{{\alpha }^{-1}}\circ S_{\alpha }\left(h\right)=g_{\alpha }h{g}_{\alpha }^{-1}$ for all $h\in H_{\alpha }$. Similarly for parts (3) and (4). Part (5) follows from the calculations below:

$$\begin{array}{cc}\hfill {\epsilon }_{{\alpha }^2}\left(g_{\alpha }\right)& ={\epsilon }_{{\alpha }^2}\left(u_{\alpha }{\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)={\epsilon }_{{\alpha }^2}\left(u_{\alpha }{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)\right)\hfill \\ \hfill & ={\epsilon }_{\alpha }\left(u_{\alpha }\right){\epsilon }_{\alpha }\left(S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)\right)={\epsilon }_{\alpha }\left(u_{\alpha }\right){\epsilon }_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)={\epsilon }_1\left(u_{\alpha }{u}_{\alpha }^{-1}\right)={\epsilon }_1\left(1\right)=1_{\mathbb{k}}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\phi }_{\beta }\left(g_{\alpha }\right)=& {\phi }_{\beta }\left(u_{\alpha }{\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)={\phi }_{\beta }\left(u_{\alpha }\right){\phi }_{\beta }\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)\hfill \\ \hfill =& {\phi }_{\beta }\left(u_{\alpha }\right){\phi }_{\beta }\left(S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)\right)={\phi }_{\beta }\left(u_{\alpha }\right)S_{\beta {\alpha }^{-1}{\beta }^{-1}}\left({\phi }_{\beta }\left(u_{\alpha }^{-1}\right)\right)\hfill \\ \hfill =& u_{\beta \alpha {\beta }^{-1}}{S}_{\beta {\alpha }^{-1}{\beta }^{-1}}\left(u_{\beta \alpha {\beta }^{-1}}^{-1}\right)=u_{\beta \alpha {\beta }^{-1}}{\left(S_{\beta \alpha {\beta }^{-1}}\left(u_{\beta \alpha {\beta }^{-1}}\right)\right)}^{-1}=g_{\beta \alpha {\beta }^{-1}}.\hfill \end{array}$$

□

Corollary 13.

$S_{{\beta }^{-1}}\circ S_{\beta }\circ S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)=u_{\alpha }{\phi }_{\beta }\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)h{\left({\phi }_{{\beta }^{-1}}\left(u_{\alpha }\right){\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)}^{-1}$ for all $\beta \in \pi$ and $h\in H_{\beta }$, where $u_{\alpha }=S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}$.

Proof. 

Observe from Propositon 4 and Corollary 9 that $u_{\alpha }\in H_{\alpha }$ is invertible and $S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)=u_{\alpha }h{\phi }_{{\beta }^{-1}}\left(u_{\alpha }^{-1}\right)={\phi }_{\beta }\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)h{S}_{\alpha }\left(u_{\alpha }\right)$ for all $h\in H_{\beta }$, then

$$\begin{array}{cc}\hfill S_{{\beta }^{-1}}& \circ S_{\beta }\circ S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)=S_{{\beta }^{-1}}\circ S_{\beta }\left(u_{\alpha }h{\phi }_{{\beta }^{-1}}\left(u_{\alpha }^{-1}\right)\right)\hfill \\ \hfill =& S_{{\alpha }^{-1}}\circ S_{\alpha }\left(u_{\alpha }\right)S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)S_{{\beta }^{-1}\alpha \beta }\circ S_{{\beta }^{-1}{\alpha }^{-1}\beta }\left({\phi }_{{\beta }^{-1}}\left(u_{\alpha }^{-1}\right)\right)\hfill \\ \hfill =& S_{{\alpha }^{-1}}\circ S_{\alpha }\left(u_{\alpha }\right)S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)S_{{\beta }^{-1}\alpha \beta }\circ S_{{\beta }^{-1}{\alpha }^{-1}\beta }\left(u_{{\beta }^{-1}\alpha \beta }^{-1}\right)\hfill \\ \hfill =& u_{\alpha }{\phi }_{\beta }\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)h{S}_{\alpha }\left(u_{\alpha }\right)u_{{\beta }^{-1}\alpha \beta }^{-1}\hfill \\ \hfill =& u_{\alpha }{\phi }_{\beta }\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)h{\left(u_{{\beta }^{-1}\alpha \beta }{\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)}^{-1}\hfill \\ \hfill =& u_{\alpha }{\phi }_{\beta }\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)h{\left({\phi }_{{\beta }^{-1}}\left(u_{\alpha }\right){\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)}^{-1},\hfill \end{array}$$

or equivalently $S_{{\beta }^{-1}}\circ S_{\beta }\circ S_{{\beta }^{-1}}\circ S_{\beta }\left(h\right)=u_{\alpha }{\phi }_{\beta }\left({\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)h{\left({\phi }_{{\beta }^{-1}}\left(u_{\alpha }\right){\left(S_{\alpha }\left(u_{\alpha }\right)\right)}^{-1}\right)}^{-1}$ for all $h\in H_{\beta }$. □

Proposition 8.

Let $(H,\mathcal{R})$ be a quasitriangular Hopf non-coassociative π-algebra with antipode S over $\mathbb{k}$ and let $u={\left\{u_{\alpha }\right\}}_{\alpha \in \pi }$ be the Drinfel’d element of $(H,\mathcal{R})$. If the second tensor factor of ${\mathcal{R}}_{1,1}$ is coassociative, then the following hold:

 (a)

${\Delta }_{\alpha }\left(u_{\alpha }\right)={\left({\sigma }_{H_1,H_1}\left({\mathcal{R}}_{1,1}\right){\mathcal{R}}_{1,{\alpha }^{-1}}\right)}^{-1}(u_{\alpha }\otimes u_1)=(u_{\alpha }\otimes u_1){\left({\sigma }_{H_1,H_1}\left({\mathcal{R}}_{1,1}\right){\mathcal{R}}_{1,{\alpha }^{-1}}\right)}^{-1}$ and ${\epsilon }_{\alpha }\left(u_{\alpha }\right)=1_{\mathbb{k}}$.

 (b)

${\Delta }_{{\alpha }^{-1}}{S}_{\alpha }\left(u_{\alpha }\right)={\left({\sigma }_{H_1,H_{\alpha }}\left({\mathcal{R}}_{1,\alpha }\right){\mathcal{R}}_{1,1}\right)}^{-1}(S_1\left(u_1\right)\otimes S_{\alpha }\left(u_{\alpha }\right))$.

 (c)

${\Delta }_{\alpha }{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)={\sigma }_{H_1,H_{\alpha }}\left({\mathcal{R}}_{1,\alpha }\right){\mathcal{R}}_{1,1}(S_1\left(u_1^{-1}\right)\otimes S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right))$.

 (d)

$g_1=u_1{\left(S_1\left(u_1\right)\right)}^{-1}$ is a group-like element of $H_1$.

Proof. 

To show part (a), we write ${\mathcal{R}}_{\alpha ,\beta }={\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}\otimes {\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}$. Therefore $u_{\alpha }=S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}$.

Applying ${\Delta }_1\otimes {\mathrm{id}}_{H_{{\alpha }^{-1}}}\otimes {\mathrm{id}}_{H_{{\alpha }^{-1}}}$ and ${\mathrm{id}}_{H_1}\otimes {\mathrm{id}}_{H_1}\otimes {\Delta }_{{\alpha }^{-1}}$ to both sides of Equation (36), respectively, we obtain

$$\begin{array}{cc}& {\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{}_{\left(1\right)}\otimes {\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{}_{\left(2\right)}\otimes {\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}{}_{\left(1\right)}\otimes {\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}{}_{\left(2\right)}\hfill \\ \hfill =& {\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{}_{\left(1\right)}{\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(1\right)}{}_{\left(1\right)}\otimes {\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{}_{\left(2\right)}{\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(1\right)}{}_{\left(2\right)}\otimes {\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(2\right)}\otimes {\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\hfill \\ \hfill =& {\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(1\right)}\otimes {\mathcal{R}}_{1,1}^{\left(1\right)}{\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}\otimes {\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(2\right)}{\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\otimes {\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(2\right)}.\hfill \end{array}$$

Using Proposition 6 and part (1) of Theoerm 11, we calculate

$$\begin{array}{cc}\hfill {\Delta }_{\alpha }\left(u_{\alpha }\right)=& {\Delta }_{\alpha }\left(S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}\right)={\Delta }_{\alpha }\left(S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right)\right){\Delta }_1\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}\right)\hfill \\ \hfill =& (S_{{\alpha }^{-1}}{\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right)}_{\left(1\right)}\otimes S_{{\alpha }^{-1}}{\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right)}_{\left(2\right)})({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{}_{\left(1\right)}\otimes {\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{}_{\left(2\right)})\hfill \\ \hfill =& (S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}{}_{\left(2\right)}\right)\otimes S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}{}_{\left(1\right)}\right))({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{}_{\left(1\right)}\otimes {\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{}_{\left(2\right)})\hfill \\ \hfill =& S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}{}_{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{}_{\left(1\right)}\otimes S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}{}_{\left(1\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{}_{\left(2\right)}\hfill \\ \hfill =& S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(1\right)}\otimes S_{{\alpha }^{-1}}\left({\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(2\right)}{\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\right){\mathcal{R}}_{1,1}^{\left(1\right)}{\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}\hfill \\ \hfill =& S_1\left({\mathcal{R}}_{1,1}^{\left(2\right)}\right)S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(1\right)}\otimes S_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\right)S_{{\alpha }^{-1}}\left({\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,1}^{\left(1\right)}{\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}\hfill \\ \hfill =& S_1\left({\mathcal{R}}_{1,1}^{\left(2\right)}\right)u_{\alpha }{\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(1\right)}\otimes S_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\right)S_{{\alpha }^{-1}}\left({\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,1}^{\left(1\right)}{\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}\hfill \\ \hfill =& S_1\left({\mathcal{R}}_{1,1}^{\left(2\right)}\right)S_1\circ S_1\left({\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(1\right)}\right)u_{\alpha }\otimes S_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\right)S_{{\alpha }^{-1}}\left({\widehat{\mathcal{R}}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,1}^{\left(1\right)}{\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}\hfill \\ \hfill =& S_1\left({\mathcal{R}}_{1,1}^{\left(2\right)}\right)S_1\left({\widehat{\mathcal{R}}}_{1,\alpha }^{\left(1\right)}\right)u_{\alpha }\otimes S_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\right){\widehat{\mathcal{R}}}_{1,\alpha }^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(1\right)}{\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}\hfill \end{array}$$

and thus

$${\Delta }_{\alpha }\left(u_{\alpha }\right)=S_1\left({\widehat{\mathcal{R}}}_{1,\alpha }^{\left(1\right)}{\mathcal{R}}_{1,1}^{\left(2\right)}\right)u_{\alpha }\otimes S_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\right){\widehat{\mathcal{R}}}_{1,\alpha }^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(1\right)}{\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}.$$

Since H is quasitriangular, S is bijective by Theorem 12. Write ${\mathcal{R}}_{\alpha ,\beta }={\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}\otimes {\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}$. By Equations (28) and (29), we have

$$\begin{array}{cc}\hfill & h_{\left(2\right)}\otimes {\Delta }_{\gamma }^{cop}\left(h_{\left(1\right)}\right)({\phi }_{{\gamma }^{-1}}\otimes {\phi }_{{\gamma }^{-1}})\left({\mathcal{R}}_{\alpha ,\beta }\right)=h_{\left(2\right)}\otimes {\mathcal{R}}_{\alpha ,\beta }{\Delta }_{\gamma }\left(h_{\left(1\right)}\right)\hfill \\ \hfill ⟹& S_{{\gamma }^{-1}}^{-1}\left(h_{\left(2\right)}\right)\otimes {\Delta }_{\gamma }^{cop}\left(h_{\left(1\right)}\right){\mathcal{R}}_{{\gamma }^{-1}\alpha \gamma ,{\gamma }^{-1}\beta \gamma }=S_{{\gamma }^{-1}}^{-1}\left(h_{\left(2\right)}\right)\otimes {\mathcal{R}}_{\alpha ,\beta }{\Delta }_{\gamma }\left(h_{\left(1\right)}\right)\hfill \\ \hfill ⟹& S_{{\gamma }^{-1}}^{-1}\left(h_{\left(2\right)}\right)h_{\left(1\right)\left(2\right)}{\mathcal{R}}_{{\gamma }^{-1}\alpha \gamma ,{\gamma }^{-1}\beta \gamma }^{\left(1\right)}\otimes h_{\left(1\right)\left(1\right)}{\mathcal{R}}_{{\gamma }^{-1}\alpha \gamma ,{\gamma }^{-1}\beta \gamma }^{\left(2\right)}=S_{{\gamma }^{-1}}^{-1}\left(h_{\left(2\right)}\right){\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}{h}_{\left(1\right)\left(1\right)}\otimes {\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}{h}_{\left(1\right)\left(2\right)}\hfill \\ \hfill ⟹& {\mathcal{R}}_{{\gamma }^{-1}\alpha \gamma ,{\gamma }^{-1}\beta \gamma }^{\left(1\right)}\otimes h{\mathcal{R}}_{{\gamma }^{-1}\alpha \gamma ,{\gamma }^{-1}\beta \gamma }^{\left(2\right)}=S_{{\gamma }^{-1}}^{-1}\left(h_{\left(2\right)}\right){\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}{h}_{\left(1\right)\left(1\right)}\otimes {\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}{h}_{\left(1\right)\left(2\right)}\hfill \\ \hfill ⟹& {\mathcal{R}}_{{\gamma }^{-1}\alpha \gamma ,{\gamma }^{-1}\beta \gamma }^{\left(1\right)}{\mathcal{R}}_{{\gamma }^{-1}\xi \gamma ,{\gamma }^{-1}\zeta \gamma }^{\left(2\right)}\otimes h{\mathcal{R}}_{{\gamma }^{-1}\alpha \gamma ,{\gamma }^{-1}\beta \gamma }^{\left(2\right)}{\mathcal{R}}_{{\gamma }^{-1}\xi \gamma ,{\gamma }^{-1}\zeta \gamma }^{\left(1\right)}\hfill \\ \hfill =& S_{{\gamma }^{-1}}^{-1}\left(h_{\left(2\right)}\right){\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}{h}_{\left(1\right)\left(1\right)}{\mathcal{R}}_{{\gamma }^{-1}\xi \gamma ,{\gamma }^{-1}\zeta \gamma }^{\left(2\right)}\otimes {\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}{h}_{\left(1\right)\left(2\right)}{\mathcal{R}}_{{\gamma }^{-1}\xi \gamma ,{\gamma }^{-1}\zeta \gamma }^{\left(1\right)},\hfill \end{array}$$

from which we derive the commutation relation:

$$\begin{array}{cc}\hfill {\mathcal{R}}_{{\gamma }^{-1}\alpha \gamma ,{\gamma }^{-1}\beta \gamma }^{\left(1\right)}& {\mathcal{R}}_{{\gamma }^{-1}\xi \gamma ,{\gamma }^{-1}\zeta \gamma }^{\left(2\right)}\otimes h{\mathcal{R}}_{{\gamma }^{-1}\alpha \gamma ,{\gamma }^{-1}\beta \gamma }^{\left(2\right)}{\mathcal{R}}_{{\gamma }^{-1}\xi \gamma ,{\gamma }^{-1}\zeta \gamma }^{\left(1\right)}\hfill \\ \hfill =& S_{{\gamma }^{-1}}^{-1}\left(h_{\left(2\right)}\right){\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}{\mathcal{R}}_{\xi ,\zeta }^{\left(2\right)}{h}_{\left(1\right)\left(2\right)}\otimes {\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}{\mathcal{R}}_{\xi ,\zeta }^{\left(1\right)}{h}_{\left(1\right)\left(1\right)}.\hfill \end{array}$$

(40)

Applying ${\mathrm{id}}_{H_{\alpha \beta }}\otimes {\Delta }_{\gamma }\otimes {\mathrm{id}}_{H_{\gamma }}$ to both sides of the equation of Equation (36) we obtain

$$\begin{array}{cc}\hfill {\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(1\right)}& \otimes {\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}{}_{\left(1\right)\left(1\right)}\otimes {\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}{}_{\left(1\right)\left(2\right)}\otimes {\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}{}_{\left(2\right)}=\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(1\right)}\otimes {\Delta }_{\gamma }{\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}{}_{\left(1\right)}\otimes {\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}{}_{\left(2\right)}\right)\hfill \\ \hfill =& \left({\mathcal{R}}_{\alpha ,\gamma }^{\left(1\right)}{\mathcal{R}}_{\beta ,\gamma }^{\left(1\right)}\otimes {\Delta }_{\gamma }{\mathcal{R}}_{\beta ,\gamma }^{\left(2\right)}\otimes {\mathcal{R}}_{\alpha ,\gamma }^{\left(2\right)}\right)={\mathcal{R}}_{\alpha ,\gamma }^{\left(1\right)}{\mathcal{R}}_{\beta ,\gamma }^{\left(1\right)}\otimes {\mathcal{R}}_{\beta ,\gamma }^{\left(2\right)}{}_{\left(1\right)}\otimes {\mathcal{R}}_{\beta ,\gamma }^{\left(2\right)}{}_{\left(2\right)}\otimes {\mathcal{R}}_{\alpha ,\gamma }^{\left(2\right)}\hfill \\ \hfill =& {\mathcal{R}}_{\alpha ,\gamma }^{\left(1\right)}{\mathcal{R}}_{\beta ,\gamma }^{\left(1\right)}{\mathcal{R}}_{1,\gamma }^{\left(1\right)}\otimes {\mathcal{R}}_{1,\gamma }^{\left(2\right)}\otimes {\mathcal{R}}_{\beta ,\gamma }^{\left(2\right)}\otimes {\mathcal{R}}_{\alpha ,\gamma }^{\left(2\right)}={\mathcal{R}}_{\alpha ,\gamma }^{\left(1\right)}{\mathcal{R}}_{1,\gamma }^{\left(1\right)}{\mathcal{R}}_{\beta ,\gamma }^{\left(1\right)}\otimes {\mathcal{R}}_{\beta ,\gamma }^{\left(2\right)}\otimes {\mathcal{R}}_{1,\gamma }^{\left(2\right)}\otimes {\mathcal{R}}_{\alpha ,\gamma }^{\left(2\right)},\hfill \end{array}$$

hence

$$\begin{array}{cc}\hfill {\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(1\right)}& \otimes S_{\gamma }{\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}\right)}_{\left(1\right)}\otimes S_{\gamma }{\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}\right)}_{\left(2\right)\left(1\right)}\otimes S_{\gamma }{\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}\right)}_{\left(2\right)\left(2\right)}\hfill \\ \hfill =& {\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(1\right)}\otimes S_{\gamma }\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}{}_{\left(2\right)}\right)\otimes S_{\gamma }{\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}{}_{\left(1\right)}\right)}_{\left(1\right)}\otimes S_{\gamma }{\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}{}_{\left(1\right)}\right)}_{\left(2\right)}\hfill \\ \hfill =& {\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(1\right)}\otimes S_{\gamma }\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}{}_{\left(2\right)}\right)\otimes S_{\gamma }\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}{}_{\left(1\right)\left(2\right)}\right)\otimes S_{\gamma }\left({\mathcal{R}}_{\alpha \beta ,\gamma }^{\left(2\right)}{}_{\left(1\right)\left(1\right)}\right)\hfill \\ \hfill =& {\mathcal{R}}_{\alpha ,\gamma }^{\left(1\right)}{\mathcal{R}}_{1,\gamma }^{\left(1\right)}{\mathcal{R}}_{\beta ,\gamma }^{\left(1\right)}\otimes S_{\gamma }\left({\mathcal{R}}_{\alpha ,\gamma }^{\left(2\right)}\right)\otimes S_{\gamma }\left({\mathcal{R}}_{1,\gamma }^{\left(2\right)}\right)\otimes S_{\gamma }\left({\mathcal{R}}_{\beta ,\gamma }^{\left(2\right)}\right).\hfill \end{array}$$

(41)

Applying ${\mathrm{id}}_{H_1}\otimes S_1$ to both sides of ${\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}{S}_{\alpha }\left({\mathcal{R}}_{\alpha ,{\beta }^{-1}}^{\left(1\right)}\right)\otimes {\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}{\mathcal{R}}_{\alpha ,{\beta }^{-1}}^{\left(2\right)}=1\otimes 1$, which follows from Theorem 10, and using part (1) of Theorem 11, we obtain

$$\begin{array}{cc}\hfill & {\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}{S}_{\alpha }\left({\mathcal{R}}_{\alpha ,{\beta }^{-1}}^{\left(1\right)}\right)\otimes S_1\left({\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}{\mathcal{R}}_{\alpha ,{\beta }^{-1}}^{\left(2\right)}\right)=1\otimes S_1\left(1\right)\hfill \\ \hfill ⟹& {\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}{S}_{\alpha }\left({\mathcal{R}}_{\alpha ,{\beta }^{-1}}^{\left(1\right)}\right)\otimes S_{{\beta }^{-1}}\left({\mathcal{R}}_{\alpha ,{\beta }^{-1}}^{\left(2\right)}\right)S_{\beta }\left({\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}\right)=1\otimes 1\hfill \\ \hfill ⟹& {\mathcal{R}}_{\alpha ,\beta }^{\left(1\right)}{\mathcal{R}}_{{\alpha }^{-1},\beta }^{\left(1\right)}\otimes {\mathcal{R}}_{{\alpha }^{-1},\beta }^{\left(2\right)}{S}_{\beta }\left({\mathcal{R}}_{\alpha ,\beta }^{\left(2\right)}\right)=1\otimes 1.\hfill \end{array}$$

(42)

Using Equations (40)–(42) as well as part (1) of Theorem 11 again, we continue our calculation of

$$\begin{array}{cc}\hfill {\Delta }_{\alpha }\left(u_{\alpha }\right)=& S_1\left({\mathcal{R}}_{1,\alpha }^{\left(1\right)}{\mathcal{R}}_{1,1}^{\left(2\right)}\right)u_{\alpha }\otimes S_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\right){\mathcal{R}}_{1,\alpha }^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(1\right)}{\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}\hfill \\ \hfill =& S_1\left(S_1^{-1}\left(S_1{\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\right)}_{\left(2\right)}\right){\mathcal{R}}_{1,\alpha }^{\left(1\right)}{\mathcal{R}}_{1,1}^{\left(2\right)}{S}_1{\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\right)}_{\left(1\right)\left(2\right)}\right)u_{\alpha }\otimes {\mathcal{R}}_{1,\alpha }^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(1\right)}{S}_1{\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\right)}_{\left(1\right)\left(1\right)}{\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}\hfill \\ \hfill =& S_1\left(S_1^{-1}\left(S_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}{}_{\left(1\right)}\right)\right){\mathcal{R}}_{1,\alpha }^{\left(1\right)}{\mathcal{R}}_{1,1}^{\left(2\right)}{S}_1{\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}{}_{\left(2\right)}\right)}_{\left(2\right)}\right)u_{\alpha }\otimes {\mathcal{R}}_{1,\alpha }^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(1\right)}{S}_1{\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}{}_{\left(2\right)}\right)}_{\left(1\right)}{\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}\hfill \\ \hfill =& S_1\left(S_1^{-1}\left(S_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}{}_{\left(1\right)}\right)\right){\mathcal{R}}_{1,\alpha }^{\left(1\right)}{\mathcal{R}}_{1,1}^{\left(2\right)}{S}_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}{}_{\left(2\right)\left(1\right)}\right)\right)u_{\alpha }\otimes {\mathcal{R}}_{1,\alpha }^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(1\right)}{S}_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}{}_{\left(2\right)\left(2\right)}\right){\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}\hfill \\ \hfill =& S_1\left(S_1^{-1}\left(S_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}{}_{\left(1\right)\left(1\right)}\right)\right){\mathcal{R}}_{1,\alpha }^{\left(1\right)}{\mathcal{R}}_{1,1}^{\left(2\right)}{S}_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}{}_{\left(1\right)\left(2\right)}\right)\right)u_{\alpha }\otimes {\mathcal{R}}_{1,\alpha }^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(1\right)}{S}_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}{}_{\left(2\right)}\right){\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}\hfill \\ \hfill =& S_1\left(S_1^{-1}\left(S_1\left({\widehat{\widehat{\widehat{\mathcal{R}}}}}_{1,1}^{\left(2\right)}\right)\right){\mathcal{R}}_{1,\alpha }^{\left(1\right)}{\mathcal{R}}_{1,1}^{\left(2\right)}{S}_1\left({\widehat{\widehat{\mathcal{R}}}}_{1,1}^{\left(2\right)}\right)\right)u_{\alpha }\otimes {\mathcal{R}}_{1,\alpha }^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(1\right)}{S}_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\right){\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}{\widehat{\widehat{\mathcal{R}}}}_{1,1}^{\left(1\right)}{\widehat{\widehat{\widehat{\mathcal{R}}}}}_{1,1}^{\left(1\right)}\hfill \\ \hfill =& S_1\left({\widehat{\widehat{\widehat{\mathcal{R}}}}}_{1,1}^{\left(2\right)}{\mathcal{R}}_{1,\alpha }^{\left(1\right)}{\mathcal{R}}_{1,1}^{\left(2\right)}{S}_1\left({\widehat{\widehat{\mathcal{R}}}}_{1,1}^{\left(2\right)}\right)\right)u_{\alpha }\otimes {\mathcal{R}}_{1,\alpha }^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(1\right)}{S}_1\left({\widehat{\mathcal{R}}}_{1,1}^{\left(2\right)}\right){\widehat{\mathcal{R}}}_{1,1}^{\left(1\right)}{\widehat{\widehat{\mathcal{R}}}}_{1,1}^{\left(1\right)}{\widehat{\widehat{\widehat{\mathcal{R}}}}}_{1,1}^{\left(1\right)}\hfill \\ \hfill =& S_1\circ S_1\left({\widehat{\widehat{\mathcal{R}}}}_{1,1}^{\left(2\right)}\right)S_1\left({\mathcal{R}}_{1,1}^{\left(2\right)}\right)S_1\left({\mathcal{R}}_{1,\alpha }^{\left(1\right)}\right)S_1\left({\widehat{\widehat{\widehat{\mathcal{R}}}}}_{1,1}^{\left(2\right)}\right)u_{\alpha }\otimes {\mathcal{R}}_{1,\alpha }^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(1\right)}{u}_1{\widehat{\widehat{\mathcal{R}}}}_{1,1}^{\left(1\right)}{\widehat{\widehat{\widehat{\mathcal{R}}}}}_{1,1}^{\left(1\right)}\hfill \end{array}$$

$$\begin{array}{cc}\hfill =& S_1\circ S_1\left({\widehat{\widehat{\mathcal{R}}}}_{1,1}^{\left(2\right)}\right)S_1\left({\mathcal{R}}_{1,1}^{\left(2\right)}\right)S_1\left({\mathcal{R}}_{1,\alpha }^{\left(1\right)}\right)S_1\left({\widehat{\widehat{\widehat{\mathcal{R}}}}}_{1,1}^{\left(2\right)}\right)u_{\alpha }\hfill \\ \hfill & \otimes {\mathcal{R}}_{1,\alpha }^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(1\right)}{S}_1\circ S_{\left(1\right)}\left({\widehat{\widehat{\mathcal{R}}}}_{1,1}^{\left(1\right)}\right)S_1\circ S_1\left({\widehat{\widehat{\widehat{\mathcal{R}}}}}_{1,1}^{\left(1\right)}\right)u_1\hfill \end{array}$$

$$\begin{array}{cc}\hfill =& {\widehat{\widehat{\mathcal{R}}}}_{1,1}^{\left(2\right)}{S}_1\left({\mathcal{R}}_{1,1}^{\left(2\right)}\right)S_1\left({\mathcal{R}}_{1,\alpha }^{\left(1\right)}\right){\widehat{\widehat{\widehat{\mathcal{R}}}}}_{1,1}^{\left(2\right)}{u}_{\alpha }\otimes {\mathcal{R}}_{1,\alpha }^{\left(2\right)}{\mathcal{R}}_{1,1}^{\left(1\right)}{\widehat{\widehat{\mathcal{R}}}}_{1,1}^{\left(1\right)}{S}_1\left({\widehat{\widehat{\widehat{\mathcal{R}}}}}_{1,1}^{\left(1\right)}\right)u_1\hfill \\ \hfill =& S_1\left({\mathcal{R}}_{1,\alpha }^{\left(1\right)}\right){\widehat{\widehat{\widehat{\mathcal{R}}}}}_{1,1}^{\left(2\right)}{u}_{\alpha }\otimes {\mathcal{R}}_{1,\alpha }^{\left(2\right)}{S}_1\left({\widehat{\widehat{\widehat{\mathcal{R}}}}}_{1,1}^{\left(1\right)}\right)u_1={\mathcal{R}}_{1,{\alpha }^{-1}}^{-1\left(1\right)}{\mathcal{R}}_{1,1}^{-1\left(2\right)}{u}_{\alpha }\otimes {\mathcal{R}}_{1,{\alpha }^{-1}}^{-1\left(2\right)}{S}_1\left({\mathcal{R}}_{1,1}^{-1\left(1\right)}\right)u_1\hfill \\ \hfill =& {\mathcal{R}}_{1,{\alpha }^{-1}}^{-1}{\sigma }_{H_1,H_1}\left({\mathcal{R}}_{1,1}^{-1}\right)(u_{\alpha }\otimes u_1)={\left({\sigma }_{H_1,H_1}\left({\mathcal{R}}_{1,1}\right){\mathcal{R}}_{1,{\alpha }^{-1}}\right)}^{-1}(u_{\alpha }\otimes u_1),\hfill \end{array}$$

from which we also have

$${\Delta }_{\alpha }\left(u_{\alpha }\right)=(u_{\alpha }\otimes u_1){\left({\sigma }_{H_1,H_1}\left({\mathcal{R}}_{1,1}\right){\mathcal{R}}_{1,{\alpha }^{-1}}\right)}^{-1}.$$

Moreover,

$$\begin{array}{cc}\hfill {\epsilon }_{\alpha }\left(u_{\alpha }\right)=& {\epsilon }_{\alpha }\left(S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}\right)={\epsilon }_{\alpha }\left(S_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right)\right){\epsilon }_1\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}\right)\hfill \\ \hfill =& {\epsilon }_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right){\epsilon }_1\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}\right)={\epsilon }_1\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}{\epsilon }_{{\alpha }^{-1}}\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}\right)\right)={\epsilon }_1\left(1\right)=1_{\mathbb{k}}.\hfill \end{array}$$

We have established part (a).

To see parts (b) and (c), we deduce from part (a) that

$$\begin{array}{cc}\hfill {\Delta }_{{\alpha }^{-1}}{S}_{\alpha }\left(u_{\alpha }\right)=& {\sigma }_{H_{{\alpha }^{-1}},H_{{\alpha }^{-1}}}(S_{\alpha }\otimes S_{\alpha }){\Delta }_{\alpha }\left(u_{\alpha }\right)\hfill \\ \hfill =& S_{\alpha }\left(u_1{\mathcal{R}}_{1,\alpha }^{\left(2\right)}{S}_1\left({\mathcal{R}}_{1,1}^{\left(1\right)}\right)\right)\otimes S_{\alpha }\left(u_{\alpha }{S}_1\left({\mathcal{R}}_{1,\alpha }^{\left(1\right)}\right){\mathcal{R}}_{1,1}^{\left(2\right)}\right)\hfill \\ \hfill =& S_1{S}_1\left({\mathcal{R}}_{1,1}^{\left(1\right)}\right)S_{\alpha }\left({\mathcal{R}}_{1,\alpha }^{\left(2\right)}\right)S_1\left(u_1\right)\otimes S_1\left({\mathcal{R}}_{1,1}^{\left(2\right)}\right)S_1{S}_1\left({\mathcal{R}}_{1,\alpha }^{\left(1\right)}\right)S_{\alpha }\left(u_{\alpha }\right)\hfill \\ \hfill =& S_1\left({\mathcal{R}}_{1,1}^{\left(1\right)}\right){\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(2\right)}{S}_1\left(u_1\right)\otimes {\mathcal{R}}_{1,1}^{\left(2\right)}{S}_1\left({\mathcal{R}}_{1,{\alpha }^{-1}}^{\left(1\right)}\right)S_{\alpha }\left(u_{\alpha }\right)\hfill \\ \hfill =& {\mathcal{R}}_{1,1}^{-1\left(1\right)}{\mathcal{R}}_{1,\alpha }^{-1\left(2\right)}{S}_1\left(u_1\right)\otimes {\mathcal{R}}_{1,1}^{-1\left(2\right)}{\mathcal{R}}_{1,\alpha }^{-1\left(1\right)}{S}_{\alpha }\left(u_{\alpha }\right)\hfill \\ \hfill =& ({\mathcal{R}}_{1,1}^{-1\left(1\right)}{\mathcal{R}}_{1,\alpha }^{-1\left(2\right)}\otimes {\mathcal{R}}_{1,1}^{-1\left(2\right)}{\mathcal{R}}_{1,\alpha }^{-1\left(1\right)})(S_1\left(u_1\right)\otimes S_{\alpha }\left(u_{\alpha }\right))\hfill \\ \hfill =& {\mathcal{R}}_{1,1}^{-1}{\sigma }_{H_1,H_{{\alpha }^{-1}}}\left({\mathcal{R}}_{1,\alpha }^{-1}\right)(S_1\left(u_1\right)\otimes S_{\alpha }\left(u_{\alpha }\right))\hfill \\ \hfill =& {\left({\sigma }_{H_1,H_{\alpha }}\left({\mathcal{R}}_{1,\alpha }\right){\mathcal{R}}_{1,1}\right)}^{-1}(S_1\left(u_1\right)\otimes S_{\alpha }\left(u_{\alpha }\right))\hfill \end{array}$$

and the two factors commute; thus

$$\begin{array}{c}\hfill {\Delta }_{\alpha }{S}_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right)={\sigma }_{H_1,H_{\alpha }}\left({\mathcal{R}}_{1,\alpha }\right){\mathcal{R}}_{1,1}(S_1\left(u_1^{-1}\right)\otimes S_{{\alpha }^{-1}}\left(u_{\alpha }^{-1}\right))\end{array}$$

and the two factors commute.

It remains to establish part (d). Consider the following calculation:

$$\begin{array}{cc}\hfill {\Delta }_1\left(g_1\right)=& {\Delta }_1\left(u_1{\left(S_1\left(u_1\right)\right)}^{-1}\right)={\Delta }_1\left(u_1\right){\Delta }_1\left({\left(S_1\left(u_1\right)\right)}^{-1}\right)={\Delta }_1\left(u_1\right){\Delta }_1\left(S_1\left(u_1^{-1}\right)\right)\hfill \\ \hfill =& (u_1\otimes u_1){\left({\sigma }_{H_1,H_1}\left({\mathcal{R}}_{1,1}\right){\mathcal{R}}_{1,1}\right)}^{-1}{\sigma }_{H_1,H_1}\left({\mathcal{R}}_{1,1}\right){\mathcal{R}}_{1,1}(S_1\left(u_1^{-1}\right)\otimes S_1\left(u_1^{-1}\right))\hfill \\ \hfill =& (u_1\otimes u_1)(S_1\left(u_1^{-1}\right)\otimes S_1\left(u_1^{-1}\right))=u_1{S}_1\left(u_1^{-1}\right)\otimes u_1{S}_1\left(u_1^{-1}\right))=g_1\otimes g_1.\hfill \end{array}$$

□

In [28], the twisting theory for quasitriangular Hopf algebras was studied by a 2-cocycle. By using the dual of cocycle (called a 2-cocycle), multiplication alteration for bialgebras was investigated in [29,30]. In what follows, we will introduce the definition of 2-cocycle for Hopf non-coassociative $\pi$-algebra.

Definition 11.

Let $(H,\phi )$ be a crossed Hopf non-coassociative π-algebra. If there exists a family $R={\{R_{\alpha ,\beta }\in H_{\alpha }\otimes H_{\beta }\}}_{\alpha ,\beta \in \pi }$ of invertible elements (the R-matrix) such that, the family R is invariant under the crossing, i.e., for any $\alpha ,\beta ,\gamma \in \pi$,

$$({\phi }_{\gamma }\otimes {\phi }_{\gamma })\left(R_{\alpha ,\beta }\right)=R_{\gamma \alpha {\gamma }^{-1},\gamma \beta {\gamma }^{-1}},$$

and, for any $\alpha ,\beta ,\gamma ,\delta \in \pi$,

$${\left(({\phi }_{\alpha }\otimes {\phi }_{\alpha })R_{\delta ,\gamma }\right)}_{12}({\Delta }_{\alpha }\otimes {\mathrm{id}}_{H_{\beta \gamma }})\left(R_{\alpha ,\beta \gamma }\right)={\left(R_{\alpha ,\beta }\right)}_{23}\left({\mathrm{id}}_{H_{\alpha \delta }}\otimes {\Delta }_{\gamma }\right)\left(R_{\alpha \delta ,\gamma }\right),$$

(43)

i.e.,

$$R_{\alpha \delta {\alpha }^{-1},\alpha \gamma {\alpha }^{-1}}^{\left(1\right)}{R}_{\alpha ,\beta \gamma }^{\left(1\right)}{}_{\left(1\right)}\otimes R_{\alpha \delta {\alpha }^{-1},\alpha \gamma {\alpha }^{-1}}^{\left(2\right)}{R}_{\alpha ,\beta \gamma }^{\left(1\right)}{}_{\left(2\right)}\otimes R_{\alpha ,\beta \gamma }^{\left(2\right)}=R_{\alpha \delta ,\gamma }^{\left(1\right)}\otimes R_{\alpha ,\beta }^{\left(1\right)}{R}_{\alpha \delta ,\gamma }^{\left(2\right)}{}_{\left(1\right)}\otimes R_{\alpha ,\beta }^{\left(2\right)}{R}_{\alpha \delta ,\gamma }^{\left(2\right)}{}_{\left(2\right)}$$

Then R is called a 2-cocycle.

From Theorem 11, it is easy to see that a quasitriangular Hopf non-coassociative $\pi$-algebra is a crossed Hopf non-coassociative $\pi$-algebra with a 2-cocycle.

Definition 12.

Let H be a Hopf non-coassociative π-algebra. We say that a family of $M={\left\{M_{\alpha }\right\}}_{\alpha \in \pi }$ is aπ-representationover H if M has a right π-module structure, it means that there is a family

$$\psi ={\{{\psi }_{\alpha ,\beta }:M_{\alpha }\otimes H_{\beta }⟶M_{\alpha \beta }\}}_{\alpha ,\beta \in \pi }$$

of $\mathbb{k}$-linear maps (the π-action), such that ψ is associative in the sense that, for any $\alpha ,\beta ,\gamma \in \pi ,$

$$\begin{array}{c}\hfill {\psi }_{\alpha \beta ,\gamma }(m_{\alpha ,\beta }\otimes i{d}_{A_{\gamma }})={\psi }_{\alpha ,\beta \gamma }(i{d}_{A_{\alpha }}\otimes {\psi }_{\beta ,\gamma });\end{array}$$

(44)

$$\begin{array}{c}\hfill {\psi }_{\alpha ,1}(i{d}_{H_{\alpha }}\otimes 1)=i{d}_{H_{\alpha }}.\end{array}$$

(45)

We shall associate with every Hopf non-coassociative $\pi$-algebra $H=(\{H_{\alpha },m,1_{\alpha }\},{\Delta }_{\alpha },$$\epsilon ,S)$ a category of $\pi$-representations $Re{p}_{\pi }\left(H\right)$ which has a natural structure of a $\pi$-category.

Explicitly, for any $\alpha \in \pi$, by an object $M_{\alpha }$ in the category $Re{p}_{\alpha }\left(H\right)$ we mean a vector space $M_{\alpha }$ is a right H-module with a structure:

$${\psi }_{\alpha }={\{{\psi }_{\alpha ,\beta }:M_{\alpha }\otimes H_{\beta }⟶M_{\alpha \beta }\}}_{\alpha ,\beta \in \pi }.$$

The category $Re{p}_{\pi }\left(H\right)$ is the disjoint union of the categories ${\left\{Re{p}_{\alpha }\right\}}_{\alpha \in \pi }$ where $Re{p}_{\alpha }\left(H\right)$ is the category of H-modules and H-linear homomorphisms. By Proposition 3, the tensor product and the unit object in $Re{p}_{\pi }\left(H\right)$ are defined in the usual way using the comultiplication ${\Delta }_H$ and the unit 1. That is,

$$h_{\alpha }·(m\otimes n)=\sum h_{\alpha \left(1\right)}·m\otimes h_{\alpha \left(2\right)}·n$$

for any $m\in M_{\beta }$ and $n\in N_{\gamma }$.

The associativity morphisms are the standard identification isomorphisms.

Furthermore, let $H=(\{H_{\alpha },m,1_{\alpha }\},{\Delta }_{\alpha },\epsilon ,S,\phi ,R)$ be a quasitriangular Hopf $\pi$-quasialgebra. The automorphism ${\phi }_{\alpha }$ of H defines an automorphism, ${\mathsf{\Phi }}_{\alpha }$ of $Re{p}_{\pi }\left(H\right)$.

If $M_{\beta }$ is in $Rep{\left(H\right)}_{\beta }$, then ${\mathsf{\Phi }}_{\alpha }\left(M\right)$ has the same underlying vector space as M and each $x\in H_{\alpha \beta {\alpha }^{-1}}$ acts as multiplication by ${\phi }_{\alpha }^{-1}\left(x\right)\in H_{\beta }$. Every $H_{\beta }$-homomorphism $M⟶N$ is mapped to itself considered as a $H_{\alpha \beta {\alpha }^{-1}}$-homomorphism. It is easy to check that $Re{p}_{\pi }\left(H\right)$ is a crossed $\pi$-category (see [4]).

A universal R-matrix $R={\{R_{\alpha ,\beta }\in H_{\alpha }\otimes H_{\beta }\}}_{\alpha ,\beta \in \pi }$ in H induces a braiding in $Re{p}_{\pi }\left(H\right)$ as follows. For $M\in Rep{\left(H\right)}_{\alpha })$ and $N\in Rep{\left(H\right)}_{\beta })$, the braiding

$$c_{M,N}:M\otimes N⟶{}^{M}N\otimes M$$

is the composition of multiplication by $R_{\alpha ,\beta }$, permutation $M\otimes N⟶N\otimes M$. The conditions defining a universal R-matrix ensure that ${\left\{c_{M,N}\right\}}_{M,N}$ is a braiding.

We now obtain

Theorem 13.

Let H be any quasitriangular Hopf non-coassociative π-algebra. Then the category $Re{p}_{\pi }\left(H\right)$ of π-representations is a braided T-category.