Group-Graded By-Product Construction and Group Double Centralizer Properties

Abstract

For a group $\pi$ with unit e, we introduce and study the notion of a $\pi$-graded Hopf algebra. Then we introduce and construct a new braided monoidal category ${}_H^{H_e}{\mathcal{YD}}_{\pi }$ over a $\pi$-graded Hopf algebra H. We introduce the notion of a $\pi$-double centralizer property and investigate this property by studying a braided $\pi$-graded Hopf algebra $U\left(g{l}_n\left(V\right)\right){⋉}_{\pi }H$, where V is an n-dimensional vector space in ${}_H^{H_e}{\mathcal{YD}}_{\pi }$ and $U\left(g{l}_n\left(V\right)\right)$ is the braided universal enveloping algebra of $g{l}_n\left(V\right)$ which is not the usual Hopf algebra. Finally, some examples and special cases are given.

1. Introduction

Braided monoidal categories have a lot of applications to pure mathematics and mathematical physics (see [1]). On the other hand, let V be a finite-dimensional vector space over a field $\mathbb{k}$ of characteristic 0. A classical theorem of Schur states that the symmetric group $S_m$ acting on $V^{\otimes m}$ and the Lie algebra $gl\left(V\right)$ acting on $V^{\otimes m}$ are precisely the centralizers of each other in $En{d}_{\mathbb{k}}{V}^{\otimes m}$. It is a powerful tool in representation theory with many applications to pure mathematics and mathematical physics (see [2]). This result was generalized by A. Berele and A. Regev [3] to the case of Lie superalgebra $pl\left(V\right)$ acting on a $Z_2$-graded vector space V.

It is now very natural to ask whether there is a new approach to constructing a symmetric braided monoidal category and to obtain a so-called group double centralizer property generalizing the classical case.

For a group $\pi$ with unit e, in this paper, we study a general $\pi$-graded Hopf algebra H. We construct new non-trivial examples of symmetric braided monoidal category ${}_H^{H_e}{\mathcal{YD}}_{\pi }$ and then establish a $\pi$-double centralizer property in the $\pi$-graded Hopf algebra framework.

The paper is organized in the following way.

In Section 2, we introduce the notion of a $\pi$-graded Hopf algebra, see Definition 1 and we study some of properties, see Theorem 1 and Proposition 1.

In Section 3, we we mainly construct a new braided monoidal category ${}_H^{H_e}\mathcal{YD}$ (different from ${}_H^H{\mathcal{YD}}_{\pi }$) of left $\pi$-graded Yetter–Drinfeld modules over a $\pi$-graded Hopf algebr H, whose morphisms are both $H_e$-comodule map and map of H-module-like objects, see Theorem 2.

In Section 4, we consider how to construct a braided $\pi$-graded Hopf algebra in ${}_H^{H_e}{\mathcal{YD}}_{\pi }$ and we get a generalized Radford $\pi$-graded by-product, see Theorems 3 and 4.

In Section 5, we introduce the notion of a $\pi$-double centralizer property, see Definition 8. Given a n-dimensional vector space V in ${}_H^{H_e}{\mathcal{YD}}_{\pi }$, we construct a braided universal enveloping algebra $U\left(g{l}_n\left(V\right)\right)$ of $g{l}_n\left(V\right)$ which is not the usual Hopf algebra, see Proposition 6. Then we investigate this property by studying a braided $\pi$-graded Hopf algebra $U\left(g{l}_n\left(V\right)\right){⋉}_{\pi }H$, see Theorem 6.

In Section 6, we consider some examples and special cases, see Theorems 7–10.

The following notations and assumptions are used.

We fix the ground field $\mathbb{k}$. Algebras, tensor products and vector spaces without indications are over $\mathbb{k}$. We denote by ${\mathbb{k}}^{\times }$ the multiplicative group of $\mathbb{k}$. For the comultiplication $\Delta$ in a co-algebra C, we use the Sweedler–Heyneman’s notation [4]: $\Delta \left(c\right)=\sum c_1\otimes c_2$, for any $c\in C$, or shortly, if we leave the summation implicit, $\Delta \left(c\right)=c_1\otimes c_2$. Similarly, for a left C-comodule $(M,{\rho }^M)$, we write ${\rho }^M\left(m\right)=m_{(-1)}\otimes m_0$ for all $m\in M$.

The material is given in the following basic references. The theory of Hopf algebras is given in [4]. A review on the theory of algebras and Hopf algebras in braided categories is given in [5]. A theory of braided monoidal category is contained in [1].

2. The Crossed $\mathbf{\pi }$-Graded Hopf Algebra

In this we will introduce the notion of a $\pi$-graded Hopf algebra and study its some of properties.

Let $\pi$ be a group with unit e. Recall that a π-graded algebra is an algebra A with the product ▽ and the unit $\eta$ such that A admits a decomposition as a direct sum of spaces $A={⨁}_{\alpha \in \pi }{A}_{\alpha }$ satisfying $A_{\alpha }{A}_{\beta }\subseteq A_{\alpha \beta }$ with $\alpha ,\beta \in \pi$ and $1:=\eta \left(1_{\mathbb{k}}\right)\in A_e$. We denote the product ▽ on $A_{\alpha }\otimes A_{\beta }$ by ${▽}_{\alpha ,\beta }$ for all $\alpha ,\beta \in \pi$.

Definition 1.

A π-graded Hopf algebra is a π-graded algebra $H={⨁}_{\alpha \in \pi }{H}_{\alpha }$ such that each $H_{\alpha }$ is a co-algebra with comultiplication ${\Delta }_{\alpha }$ and co-unit ${\epsilon }_{\alpha }$; the map $\eta :\mathbb{k}⟶A_e$ and the maps ${▽}_{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶H_{\alpha \beta }$ (for all $\alpha ,\beta \in \pi$) are co-algebra maps (we sometime write $m_{\alpha ,\beta }$ for ${▽}_{\alpha ,\beta }$), with a family $S={\{S_{\alpha }:H_{\alpha }⟶H_{{\alpha }^{-1}}\}}_{\alpha \in \pi }$ of $\mathbb{k}$-linear maps (the antipode) such that for any $\alpha \in \pi$,

$${▽}_{{\alpha }^{-1},\alpha }(S_{\alpha }\otimes {\iota }_{H_{\alpha }}){\Delta }_{\alpha }={\epsilon }_{\alpha }1={▽}_{\alpha ,{\alpha }^{-1}}({\iota }_{H_{\alpha }}\otimes S_{\alpha }){\Delta }_{\alpha }.$$

(1)

Furthermore, A crossed π-graded Hopf algebra is a π-graded Hopf algebra with a set of co-algebra isomorphism $\psi ={\{{\psi }_{\beta }:H_{\alpha }⟶H_{\beta \alpha {\beta }^{-1}}\}}_{\alpha ,\beta \in \pi }$, called crossing, satisfying the following conditions: for any $\alpha ,\beta ,\gamma \in \pi$,

(a) ψ is multiplicative, i.e., ${\psi }_{\alpha }\circ {\psi }_{\beta }={\psi }_{\alpha \beta }$. It follows that ${\psi }_1\left(H_{\alpha }\right)=i{d}_{H_{\alpha }}$.

(b) ψ is compatible with ▽, i.e., ${▽}_{\gamma \alpha {\gamma }^{-1},\gamma \beta {\gamma }^{-1}}({\psi }_{\gamma }\otimes {\psi }_{\gamma })={\psi }_{\gamma }{▽}_{\alpha ,\beta }$,

(c) ψ is compatible with η, i.e., $\eta ={\psi }_{\gamma }\circ \eta$.

(d) ψ preserves the antipode, i.e., ${\psi }_{\beta }{S}_{\alpha }=S_{\beta \alpha {\beta }^{-1}}{\psi }_{\beta }$.

Example 1. (1) The group algebra $A=\mathbb{k}\left[\pi \right]$ is π-graded Hopf algebra, with $A_{\alpha }=\mathbb{k}\alpha$ for all $\alpha \in \pi$;

(2) For any Hopf algebra A, we have a π-graded Hopf algebra $H=A\left[\pi \right]$ with $H_{\alpha }=A\alpha$ for all $\alpha \in \pi$. Product in $A\left[\pi \right]$ is given by $\left(a\alpha \right)\left(b\beta \right)=\left(ab\right)\left(\alpha \beta \right)$ and coproduct on $A\left[\pi \right]$ is given $\Delta \left(a\alpha \right)=\Delta \left(a\right)(\alpha \otimes \alpha )$ for any $a,b\in A$ and $\alpha ,\beta \in \pi$;

(3) If π is a finite group, there is a one-to-one correspondence between (isomorphic classes of) π-algebras (cf. [6,7]) and (isomorphic classes of) π-graded algebras. In particular, if $A={(\left\{A_{\alpha }\right\},m,\eta )}_{\alpha \in \pi }$ is a π-algebra, then $\tilde{A}=\underset{\alpha \in \pi }{⨁}{A}_{\alpha }$ is a π-graded algebra with multiplication $\tilde{m}$ and unit $\tilde{\eta }$ given on the summands by

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

Furthermore, if $A={(\{A_{\alpha },{\Delta }_{\alpha },{\epsilon }_{\alpha }\},m,1,S,\psi )}_{\alpha \in \pi }$ is a crossed π-graded Hopf algebra, where π is a finite group. Then the algebra $(\tilde{A},\tilde{m},\tilde{\eta }),$ defined as above, is a crossed Hopf π-graded algebra with comultiplication $\tilde{\Delta }$, co-unit element $\tilde{\epsilon }$, and antipode $\tilde{S}$ given by

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

(4) Let $H=(\{H_{\alpha },m_{\alpha },1_{\alpha }\},\Delta ,\epsilon ,S,\phi )$ be a finite type crossed Hopf π-co-algebra with crossing φ (cf. [8]) (or Turaev’s group co-algebra (cf. [9]). Then the π-graded algebra $H^{\ast }={⨁}_{\alpha \in \pi }{H}_{\alpha }^{\ast }$ inherits a structure of a crossed π-graded Hopf algebra by setting, for all $\alpha \in \pi$ and $x\in H_{\alpha }^{\ast }$,

$${\Delta }_{\alpha }\left(x\right)=m_{\alpha }^{\ast }\left(x\right),{\epsilon }_{\alpha }\left(x\right)=x\left(1_{\alpha }\right),S\left(x\right)=x\circ S_{{\alpha }^{-1}},and{\psi }_{\alpha }={\phi }_{{\alpha }^{-1}}^{\ast }.$$

We have the following main result of this section.

Theorem 1.

Let $H={⨁}_{\alpha \in \pi }{H}_{\alpha }$ be a π-graded Hopf algebra with antipode S. Then

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

(2)

$$\begin{array}{c}{S}_1\left(1\right)=1;\end{array}$$

(3)

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

(4)

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

(5)

Proof. 

We compute as follows:

$$\begin{array}{cc}\hfill & S_{\alpha \beta }\left(ab\right)=S_{\alpha \beta }{▽}_{\alpha ,\beta }(a\otimes b)=S_{\alpha \beta }{▽}_{\alpha ,\beta }{\sigma }_{H_{\beta },H_{\alpha }}(b\otimes a)\hfill \\ \hfill =& 1S_{\alpha \beta }{▽}_{\alpha ,\beta }{\sigma }_{H_{\beta },H_{\alpha }}(b\otimes a)={▽}_{1,{\beta }^{-1}{\alpha }^{-1}}(1\otimes S_{\alpha \beta }{▽}_{\alpha ,\beta }{\sigma }_{H_{\beta },H_{\alpha }}(b\otimes a))\hfill \\ \hfill =& {▽}_{1,{\beta }^{-1}{\alpha }^{-1}}(i{d}_{H_1}\otimes S_{\alpha \beta }{▽}_{\alpha ,\beta }{\sigma }_{H_{\beta },H_{\alpha }})(1\otimes b\otimes a)\hfill \\ \hfill =& {▽}_{1,{\beta }^{-1}{\alpha }^{-1}}(i{d}_{H_1}\otimes S_{\alpha \beta }{▽}_{\alpha ,\beta }{\sigma }_{H_{\beta },H_{\alpha }})(S_{\beta }\left(b_{\left(1\right)}\right)b_{\left(2\right)}\otimes b_{\left(3\right)}\otimes a)\hfill \\ \hfill =& {▽}_{1,{\beta }^{-1}{\alpha }^{-1}}(S_{\beta }\left(b_{\left(1\right)}\right)b_{\left(2\right)}\otimes S_{\alpha \beta }{▽}_{\alpha ,\beta }{\sigma }_{H_{\beta },H_{\alpha }}(b_{\left(3\right)}\otimes a))\hfill \\ \hfill =& {▽}_{1,{\beta }^{-1}{\alpha }^{-1}}(S_{\beta }\left(b_{\left(1\right)}\right)b_{\left(2\right)}\otimes S_{\alpha \beta }{▽}_{\alpha ,\beta }(a\otimes b_{\left(3\right)}))\hfill \\ \hfill =& {▽}_{1,{\beta }^{-1}{\alpha }^{-1}}(S_{\beta }\left(b_{\left(1\right)}\right)b_{\left(2\right)}\otimes S_{\alpha \beta }\left(a{b}_{\left(3\right)}\right))\hfill \\ \hfill =& \left(S_{\beta }\left(b_{\left(1\right)}\right)b_{\left(2\right)}\right)S_{\alpha \beta }\left(a{b}_{\left(3\right)}\right)=\left(S_{\beta }\left(b_{\left(1\right)}\right)1b_{\left(2\right)}\right)S_{\alpha \beta }\left(a{b}_{\left(3\right)}\right)\hfill \\ \hfill =& {▽}_{1,{\beta }^{-1}{\alpha }^{-1}}({▽}_{{\beta }^{-1},1,\beta }(S_{\beta }\left(b_{\left(1\right)}\right)\otimes 1\otimes b_{\left(2\right)})\otimes S_{\alpha \beta }{▽}_{\alpha ,\beta }(a\otimes b_{\left(3\right)}))\hfill \\ \hfill =& {▽}_{1,{\beta }^{-1}{\alpha }^{-1}}({▽}_{{\beta }^{-1},1,\beta }\otimes S_{\alpha \beta }{▽}_{\alpha ,\beta })(S_{\beta }\left(b_{\left(1\right)}\right)\otimes 1\otimes b_{\left(2\right)}\otimes a\otimes b_{\left(3\right)}))\hfill \\ \hfill =& {▽}_{1,{\beta }^{-1}{\alpha }^{-1}}({▽}_{{\beta }^{-1},1,\beta }\otimes S_{\alpha \beta }{▽}_{\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)}\otimes b_{\left(3\right)}))\hfill \\ \hfill =& {▽}_{1,{\beta }^{-1}{\alpha }^{-1}}({▽}_{{\beta }^{-1},1,\beta }\otimes S_{\alpha \beta }{▽}_{\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)}\otimes a_{\left(3\right)}\otimes b_{\left(2\right)}\otimes b_{\left(3\right)}))\hfill \\ \hfill =& {▽}_{1,{\beta }^{-1}{\alpha }^{-1}}({▽}_{{\beta }^{-1},1,\beta }\otimes S_{\alpha \beta }{▽}_{\alpha ,\beta })(S_{\beta }\left(b_{\left(1\right)}\right)\otimes S_{\alpha }\left(a_{\left(1\right)}\right)a_{\left(2\right)}\otimes b_{\left(2\right)}\otimes a_{\left(3\right)}\otimes b_{\left(3\right)}))\hfill \\ \hfill =& {▽}_{1,{\beta }^{-1}{\alpha }^{-1}}({▽}_{{\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)}\otimes b_{\left(2\right)})\otimes S_{\alpha \beta }{▽}_{\alpha ,\beta }(a_{\left(3\right)}\otimes b_{\left(3\right)})))\hfill \\ \hfill =& {▽}_{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)}\right)b_{\left(2\right)}\otimes S_{\alpha \beta }\left(a_{\left(3\right)}{b}_{\left(3\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)}\right)b_{\left(2\right)}\right)S_{\alpha \beta }\left(a_{\left(3\right)}{b}_{\left(3\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)}{b}_{\left(2\right)}{S}_{\alpha \beta }\left(a_{\left(3\right)}{b}_{\left(3\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 }_{\alpha \beta }\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 }_{\alpha }\left(a_{\left(2\right)}\right){\epsilon }_{\beta }\left(b_{\left(2\right)}\right)\hfill \\ \hfill =& S_{\beta }\left(b_{\left(1\right)}{\epsilon }_{\beta }\left(b_{\left(2\right)}\right)\right)S_{\alpha }\left(a_{\left(1\right)}{\epsilon }_{\alpha }\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 (4), for all, $\alpha \in \pi ,h\in H_{\alpha }$, we have that

$$\begin{array}{cc}& (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 \\ & (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 \\ & (S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)}\right)\otimes {\Delta }_1\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 \\ & (S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)}\right)\otimes {\Delta }_1\left(h_{\left(3\right)}{S}_{\alpha }\left(h_{\left(4\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 \\ & (S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)}\right)\otimes {\Delta }_{\alpha }\left(h_{\left(3\right)}\right){\Delta }_{{\alpha }^{-1}}{S}_{\alpha }\left(h_{\left(4\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 \\ & (S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)}\right)\otimes (h_{\left(3\right)}\otimes h_{\left(4\right)})(S_{\alpha }{\left(h_{\left(5\right)}\right)}_{\left(1\right)}\otimes S_{\alpha }{\left(h_{\left(5\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 \\ & (S_{\alpha }\left(h_{\left(1\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)}\right)\otimes h_{\left(3\right)}{S}_{\alpha }{\left(h_{\left(5\right)}\right)}_{\left(1\right)}\otimes h_{\left(4\right)}{S}_{\alpha }{\left(h_{\left(5\right)}\right)}_{\left(2\right)})\hfill \\ \hfill =& S_{\alpha }\left(h_{\left(1\right)}\right)\left(h_{\left(4\right)}{S}_{\alpha }{\left(h_{\left(5\right)}\right)}_{\left(2\right)}\right)\otimes S_{\alpha }\left(h_{\left(2\right)}\right)\left(h_{\left(3\right)}{S}_{\alpha }{\left(h_{\left(5\right)}\right)}_{\left(1\right)}\right)\hfill \\ \hfill =& S_{\alpha }\left(h_{\left(1\right)}\right)\left(h_{\left(4\right)}{S}_{\alpha }{\left(h_{\left(5\right)}\right)}_{\left(2\right)}\right)\otimes \left(S_{\alpha }\left(h_{\left(2\right)}\right)h_{\left(3\right)}\right)S_{\alpha }{\left(h_{\left(5\right)}\right)}_{\left(1\right)}\hfill \\ \hfill =& S_{\alpha }\left(h_{\left(1\right)}\right)\left(h_{\left(2\right)}{S}_{\alpha }{\left(h_{\left(3\right)}\right)}_{\left(2\right)}\right)\otimes 1S_{\alpha }{\left(h_{\left(3\right)}\right)}_{\left(1\right)}\hfill \\ \hfill =& \left(S_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)}\right)S_{\alpha }{\left(h_{\left(3\right)}\right)}_{\left(2\right)}\otimes 1S_{\alpha }{\left(h_{\left(3\right)}\right)}_{\left(1\right)}\hfill \\ \hfill =& (S_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)}\otimes 1)(S_{\alpha }{\left(h_{\left(3\right)}\right)}_{\left(2\right)}\otimes S_{\alpha }{\left(h_{\left(3\right)}\right)}_{\left(1\right)})\hfill \\ \hfill =& (S_{\alpha }\left(h_{\left(1\right)}\right)h_{\left(2\right)}\otimes 1){\sigma }_{H_{{\alpha }^{-1}},H_{{\alpha }^{-1}}}{\Delta }_{{\alpha }^{-1}}{S}_{\alpha }\left(h_{\left(3\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, ${\Delta }_{{\alpha }^{-1}}{S}_{\alpha }={\sigma }_{H_{{\alpha }^{-1}},H_{{\alpha }^{-1}}}(S_{\alpha }\otimes S_{\alpha }){\Delta }_{\alpha },\forall \alpha \in \pi$.

It is easy to get (3) by (1):

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

We can obtain (5) also by (1): $\forall \alpha \in \pi ,h\in H_{\alpha },$

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

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

Proposition 1.

Let $H={⨁}_{\alpha \in \pi }{H}_{\alpha }$ be a π-graded Hopf algebra with antipode S. Then the following three statements are equivalent:

(a) 

$S_{{\alpha }^{-1}}\circ S_{\alpha }={\mathrm{id}}_{H_{\alpha }}$, for all $\alpha \in \pi$;

(b) 

$S_{\alpha }\left(a_{(2,\alpha )}\right)a_{(1,\alpha )}={\epsilon }_{\alpha }\left(a\right)1$, for all $\alpha \in \pi$ and $a\in H_{\alpha }$;

(c) 

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

Proof. 

Let us prove that Proposition 1(b) implies Proposition 1(a). By uniqueness of the right inverse, it is enough to show that, for each $\alpha \in \pi$, $S_{{\alpha }^{-1}}\circ S_{\alpha }$ is a right inverse of $S_{\alpha }$ for the convolution in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$, just as is $i{d}_{H_{\alpha }}$. Now, using Theorem 1, we get that, for all $a\in H_{\alpha }$,

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

This implies that $S_{\alpha }\ast S_{{\alpha }^{-1}}\circ S_{\alpha }=\eta {\epsilon }_{\alpha }$, hence $S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$.

Let us prove the converse implication: if $S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$ for all $\alpha \in \pi$, we have:

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

It remains to prove that Proposition 1 is equivalent to Proposition 3 to complete our proof.

If Proposition 1 holds, i.e., $S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$ for all $\alpha \in \pi$, we get:

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

Now suppose that Proposition 1(c) holds, by uniqueness of the left inverse, it is enough to show that, for each $\alpha \in \pi$, $S_{{\alpha }^{-1}}\circ S_{\alpha }$ is a left inverse of $S_{\alpha }$ for the convolution in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$, just as is $i{d}_{H_{\alpha }}$. Now, using Theorem 1, we get that, for all $a\in H_{\alpha }$,

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

This implies that $S_{{\alpha }^{-1}}\circ S_{\alpha }\ast S_{\alpha }=\eta {\epsilon }_{\alpha }$, hence $S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$. □

By looking into the proof for Proposition 1, we obtain the more precise description as follows:

Corollary 1.

Let $H={⨁}_{\alpha \in \pi }{H}_{\alpha }$ be a π-graded Hopf algebra with antipode S. Fix $\alpha \in \pi$. The following three statements hold:

(a) 

If $S_1^2=i{d}_{H_1}$ and $S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$, then $S_{\alpha }\left(a_{(2,\alpha )}\right)a_{(1,\alpha )}={\epsilon }_{\alpha }\left(a\right)1$ and $a_{(2,\alpha )}{S}_{\alpha }\left(a_{(1,\alpha )}\right)={\epsilon }_{\alpha }\left(a\right)1$;

(b) 

If $S_{\alpha }\left(a_{(2,\alpha )}\right)a_{(1,\alpha )}={\epsilon }_{\alpha }\left(a\right)1$, then $S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$;

(c) 

If $a_{(2,\alpha )}{S}_{\alpha }\left(a_{(1,\alpha )}\right)={\epsilon }_{\alpha }\left(a\right)1$, then $S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$.

Proof. 

Corollary 1(a): For all $a\in H_{\alpha }$, we get that:

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

and that

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

Corollary 1(b): By uniqueness of the right inverse, it is enough to show that, for each $\alpha \in \pi$, $S_{{\alpha }^{-1}}\circ S_{\alpha }$ is a right inverse of $S_{\alpha }$ for the convolution in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$, just as is $i{d}_{H_{\alpha }}$. Now, using Theorem 1, we get that, for all $a\in H_{\alpha }$,

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

This implies that $S_{\alpha }\ast S_{{\alpha }^{-1}}\circ S_{\alpha }=\eta {\epsilon }_{\alpha }$, hence $S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$.

Corollary 1(c): By uniqueness of the left inverse, it is enough to show that, for each $\alpha \in \pi$, $S_{{\alpha }^{-1}}\circ S_{\alpha }$ is a left inverse of $S_{\alpha }$ for the convolution in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$, just as is $i{d}_{H_{\alpha }}$. Now, using Theorem 1, we get that, for all $a\in H_{\alpha }$,

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

This implies that $S_{{\alpha }^{-1}}\circ S_{\alpha }\ast S_{\alpha }=\eta {\epsilon }_{\alpha }$, hence $S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$. □

Corollary 2.

Let $H={⨁}_{\alpha \in \pi }{H}_{\alpha }$ be a crossed π-graded Hopf algebra with antipode S and a crossing ψ. Fix $\alpha \in \pi$. The following three statements hold:

(a) 

If $S_1^2=i{d}_{H_1}$ and ${\psi }_{{\alpha }^{-1}}\circ S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$, then ${\psi }_{{\alpha }^{-1}}\circ S_{\alpha }\left(a_{(2,\alpha )}\right)a_{(1,\alpha )}={\epsilon }_{\alpha }\left(a\right)1$ and $a_{(2,\alpha )}{\psi }_{{\alpha }^{-1}}\circ S_{\alpha }\left(a_{(1,\alpha )}\right)={\epsilon }_{\alpha }\left(a\right)1$;

(b) 

If ${\psi }_{{\alpha }^{-1}}\circ S_{\alpha }\left(a_{(2,\alpha )}\right)a_{(1,\alpha )}={\epsilon }_{\alpha }\left(a\right)1$, then ${\psi }_{{\alpha }^{-1}}\circ S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$;

(c) 

If $a_{(2,\alpha )}{\psi }_{{\alpha }^{-1}}\circ S_{\alpha }\left(a_{(1,\alpha )}\right)={\epsilon }_{\alpha }\left(a\right)1$, then ${\psi }_{{\alpha }^{-1}}\circ S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$.

Proof. 

Corollary 2(a): For all $a\in H_{\alpha }$, we get that:

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

and that

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

Corollary 2(b): By the uniqueness of the right inverse, it is enough to show that, for each $\alpha \in \pi$, ${\psi }_{{\alpha }^{-1}}\circ S_{{\alpha }^{-1}}\circ S_{\alpha }$ is a right inverse of $S_{\alpha }$ for the convolution in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$, just as is $i{d}_{H_{\alpha }}$. Now, using Theorem 1, we get that, for all $a\in H_{\alpha }$:

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

This implies that $S_{\alpha }\ast {\psi }_{{\alpha }^{-1}}\circ S_{{\alpha }^{-1}}\circ S_{\alpha }=\eta {\epsilon }_{\alpha }$, hence ${\psi }_{{\alpha }^{-1}}\circ S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$.

Corollary 2(c): By uniqueness of the left inverse, it is enough to show that, for each $\alpha \in \pi$, ${\psi }_{{\alpha }^{-1}}\circ S_{{\alpha }^{-1}}\circ S_{\alpha }$ is a left inverse of $S_{\alpha }$ for the convolution in the convolution algebra $\mathrm{Conv}(H_{\alpha },H)$, just as is $i{d}_{H_{\alpha }}$. Now, using Theorem 1, we get that, for all $a\in H_{\alpha }$,

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

This implies that ${\psi }_{{\alpha }^{-1}}\circ S_{{\alpha }^{-1}}\circ S_{\alpha }\ast S_{\alpha }=\eta {\epsilon }_{\alpha }$, hence ${\psi }_{{\alpha }^{-1}}\circ S_{{\alpha }^{-1}}\circ S_{\alpha }=i{d}_{H_{\alpha }}$. □

3. A New Braided Monoidal Category ${}_{\mathbf{H}}^{{\mathbf{H}}_{\mathbf{e}}}\mathcal{YD}$

Let H be a $\pi$-graded Hopf algebra. In what follows, we will introduce the $\pi$-graded Yetter–Drinfeld category ${}_H^{H_e}\mathcal{YD}$ over H and make it a new braided monoidal category.

Definition 2.

Let $H={⨁}_{\alpha \in \pi }{H}_{\alpha }$ be a π-graded algebra and V a vector space. A left π-graded H-module-like object is a couple $V=(V,▹={\left\{{▹}_{\lambda }\right\}}_{\lambda \in \pi })$, where for any $\lambda \in \pi$, ${▹}_{\lambda }:H_{\lambda }\otimes V⟶V$ is a linear map, which will be called a module-like structure (simply, π-graded action) and denoted by ${▹}_{\lambda }(a\otimes v)=a{▹}_{\lambda }v$, such that $1{▹}_{e}v=v$ and $\left(ab\right){▹}_{\alpha \beta }v=a{▹}_{\alpha }\left(b{▹}_{\beta }v\right)$ for any $a\in H_{\alpha },b\in H_{\beta }$ with $\alpha ,\beta \in \pi$ and $v\in V$.

Given two left $\pi$-graded H-module-like objects $(V,▹)$ and $(W,{▹}^{\prime })$, a homomorphism from V to W is a linear map $f:V⟶W$ such that $f\left(a{▹}_{\alpha }v\right)=a{▹}_{\alpha }^{\prime }f\left(v\right)$, for any $a\in A_{\alpha },v\in V$ and $w\in W$ with $\alpha \in \pi$.

Definition 3.

Let H be a π-graded Hopf algebra. A left π-graded Yetter–Drinfeld module M over H is a triple $(M,▹,{\rho }_M)$ where M is both a left π-graded H-module-like object with a π-graded action $▹={\left\{{▹}_{\alpha }\right\}}_{\alpha \in \pi }$ and a left $H_e$-comodule with structure ${\rho }_M$ (we write ${\rho }_M\left(m\right):=m_{(-1,e)}\otimes m_{(0,0)})$ for any $m\in M$, satisfying the following compatibility condition:

$$a_{(1,\alpha )}{m}_{(-1,e)}\otimes a_{(2,\alpha )}{▹}_{\alpha }{m}_{(0,0)}={\left(a_{(1,\alpha )}{▹}_{\alpha }m\right)}_{(-1,e)}{a}_{(2,\alpha )}\otimes {\left(a_{(1,\alpha )}{▹}_{\alpha }m\right)}_{(0,0)}$$

(6)

for any $a\in H_{\alpha },m\in M$ with $\alpha \in \pi$.

As the usual cases, we can prove (6) is equivalent to the following:

$${\rho }_M\left(a{▹}_{\alpha }m\right)=a_{(1,\alpha )}{m}_{(-1,e)}{S}_{\alpha }\left(a_{(3,\alpha )}\right)\otimes a_{(2,\alpha )}{▹}_{\alpha }{m}_{(0,0)}$$

(7)

for any $a\in H_{\alpha },m\in M$ with $\alpha \in \pi$.

We denote ${}_H^{H_e}\mathcal{YD}$ by the category of left $\pi$-graded Yetter–Drinfeld modules over H whose morphisms consist of both $H_e$-comodule map and map of H-module-like objects.

Example 2.

Let H be a π-graded Hopf algebra. Consider $H_e$ and define:

$$a{▹}_{\lambda }x=a_{(1,\lambda )}x{S}_{\lambda }\left(a_{(2,\lambda )}\right)$$

for any $a\in H_{\lambda },x\in H_e$ and $\lambda \in \pi$. It is straightforward to check that $(H_e,{\left\{{▹}_{\lambda }\right\}}_{\lambda \in \pi },{\Delta }_1)$ is a left π-graded Yetter–Drinfeld module over H.

Example 3.

Let H be a π-graded Hopf algebra. Let $V\in {}_H^{H_e}\mathcal{YD}$ be a finite-dimensional. We consider $End\left(V\right)$ of V. Let $f\in End\left(V\right),a\in H_{\alpha }$, and $v\in V$. Define a left H-module-like object $End\left(V\right)$ and a left $H_e$-comodule on $End\left(V\right)$ by, respectively,

$$\begin{array}{cc}\hfill \left(a{▹}_{\alpha }f\right)\left(v\right)& =a_{(1,\alpha )}▹f\left(S_{\alpha }\left(a_{(2,\alpha )}\right){▹}_{{\alpha }^{-1}}v\right);\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \rho \left(f\right)\left(v\right)& ={\left(f\left(v_{(0,0)}\right)\right)}_{(-1,e)}{S}_1^{-1}\left(v_{(-1,e)}\right)\otimes {\left(f\left(v_{(0,0)}\right)\right)}_{(0,0)}.\hfill \end{array}$$

(9)

Thus, it is straightforward that $End\left(V\right)$ is a left H-module-like object and a left $H_e$-comodule.

We have a natural module action of $End\left(V\right)$ on V as $f\left(v\right)$. Then it is a morphism in ${}_H^{H_e}\mathcal{YD}$. Moreover,

$$\rho \left(f\left(v\right)\right)=f_{(-1,e)}{v}_{(-1,e)}\otimes f_{(0,0)}\left(v_{(0,0)}\right)$$

(10)

for all $f\in End\left(V\right)$ and $v\in V$.

Then by (8)–(10), it is not hard to obtain that $En{d}_{\mathbb{k}}V$ is an object in ${}_A^{A_1}\mathcal{YD}$.

The main result of this section is as follows.

Theorem 2.

With the above notations. Then the category ${}_H^{H_e}\mathcal{YD}$ of left π-graded Yetter–Drinfeld modules over H is a braided monoidal category with the following structures, for any $m\in M\in {}_H^{H_e}\mathcal{YD}$, $n\in N\in {}_H^{H_e}\mathcal{YD},a\in A_{\lambda }$ with $\lambda \in \pi$:

$$\left\{\begin{array}{c}a{▹}_{\lambda }(m\otimes n)=a_{(1,\lambda )}{▹}_{\lambda }m\otimes a_{(2,\lambda )}{▹}_{\lambda }n;\hfill \\ \rho (m\otimes n)=m_{(-1,e)}{n}_{(-1,e)}\otimes m_{(0,0)}\otimes n_{(0,0)};\hfill \\ {\tau }_{M,N}(m\otimes n)=\left(m_{(-1,e)}{▹}_{\lambda }n\right)\otimes m_{(0,0)}\hfill \end{array}\right.$$

Proof. 

First, we prove that $M\otimes N$ is a $\pi$-graded H-module-like object. It is easy to get

$$\begin{array}{cc}\hfill a{▹}_{\alpha }\left(b{▹}_{\beta }(m\otimes n)\right)=& a{▹}_{\alpha }(b_{(1,\beta )}{▹}_{\beta }m\otimes b_{(2,\beta )}{▹}_{\beta }n)\hfill \\ \hfill =& a_{(1,\alpha )}{▹}_{\alpha }\left(b_{(1,\beta )}{▹}_{\beta }m\right)\otimes a_{(2,\alpha )}{▹}_{\alpha }\left(b_{(2,\beta )}{▹}_{\beta }n\right)\hfill \\ \hfill =& \left(a_{(1,\alpha )}{b}_{(1,\beta )}\right){▹}_{\alpha \beta }m\otimes \left(a_{(2,\alpha )}{b}_{(2,\beta )}\right){▹}_{\alpha \beta }n\hfill \\ \hfill =& {\left(ab\right)}_{(1,\alpha \beta )}{▹}_{\alpha \beta }m\otimes {\left(ab\right)}_{(2,\alpha \beta )}{▹}_{\alpha \beta }n\hfill \\ \hfill =& \left(ab\right){▹}_{\alpha \beta }(m\otimes n)\hfill \end{array}$$

and

$$1{▹}_e(m\otimes n)=1{▹}_{e}m\otimes 1{▹}_{e}n=m\otimes n.$$

Second, we check that $M\otimes N$ is an $H_1$-comodule as follows:

$$\begin{array}{cc}\hfill (i{d}_{H_e}\otimes \rho )\rho (m\otimes n)=& (i{d}_{H_1}\otimes \rho )(m_{(-1,e)}{n}_{(-1,e)}\otimes m_{(0,0)}\otimes n_{(0,0)})\hfill \\ \hfill =& m_{(-1,e)}{n}_{(-1,e)}\otimes \rho (m_{(0,0)}\otimes n_{(0,0)})\hfill \\ \hfill =& m_{(-1,e)}{n}_{(-1,e)}\otimes m_{(0,0)(-1,e)}{n}_{(0,0)(-1,e)}\otimes m_{(0,0)(0,0)}\otimes n_{(0,0)(0,0)}\hfill \\ \hfill =& m_{(-1,e)(1,e)}{n}_{(-1,e)(1,e)}\otimes m_{(-1,e)(2,e)}{n}_{(-1,e)(2,e)}\otimes m_{(0,0)}\otimes n_{(0,0)}\hfill \\ \hfill =& {\Delta }_1\left(m_{(-1,e)}\right){\Delta }_1\left(n_{(-1,e)}\right)\otimes m_{(0,0)}\otimes n_{(0,0)}\hfill \\ \hfill =& {\Delta }_1\left(m_{(-1,e)}{n}_{(-1,e)}\right)\otimes m_{(0,0)}\otimes n_{(0,0)}\hfill \\ \hfill =& ({\Delta }_1\otimes i{d}_{M\otimes N})(m_{(-1,e)}{n}_{(-1,e)}\otimes m_{(0,0)}\otimes n_{(0,0)})\hfill \\ \hfill =& ({\Delta }_1\otimes i{d}_{M\otimes N})\rho (m\otimes n)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill ({\epsilon }_1\otimes i{d}_{M\otimes N})\rho (m\otimes n)=& ({\epsilon }_1\otimes i{d}_{M\otimes N})(m_{(-1,e)}{n}_{(-1,e)}\otimes m_{(0,0)}\otimes n_{(0,0)})\hfill \\ \hfill =& {\epsilon }_1\left(m_{(-1,e)}{n}_{(-1,e)}\right)m_{(0,0)}\otimes n_{(0,0)}\hfill \\ \hfill =& {\epsilon }_1\left(m_{(-1,e)}\right)m_{(0,0)}\otimes {\epsilon }_1\left(n_{(-1,e)}\right)n_{(0,0)}=m\otimes n.\hfill \end{array}$$

Third, $M\otimes N$ satisfies the compatibility condition:

$$\begin{array}{cc}\hfill & {\rho }_{M\otimes N}\left(a{▹}_{\alpha }(m\otimes n)\right)={\rho }_{M\otimes N}(a_{(1,\alpha )}{▹}_{\alpha }m\otimes a_{(2,\alpha )}{▹}_{\alpha }n)\hfill \\ \hfill =& {\left(a_{(1,\alpha )}{▹}_{\alpha }m\right)}_{(-1,e)}{\left(a_{(2,\alpha )}{▹}_{\alpha }n\right)}_{(-1,e)}\otimes {\left(a_{(1,\alpha )}{▹}_{\alpha }m\right)}_{(0,0)}\otimes {\left(a_{(2,\alpha )}{▹}_{\alpha }n\right)}_{(0,0)}\hfill \\ \hfill =& a_{(1,\alpha )(1,\alpha )}{m}_{(-1,e)}{S}_{\alpha }\left(a_{(1,\alpha )(3,\alpha )}\right)a_{(2,\alpha )(1,\alpha )}{n}_{(-1,e)}{S}_{\alpha }\left(a_{(2,\alpha )(3,\alpha )}\right)\hfill \\ & \otimes a_{(1,\alpha )(2,\alpha )}{▹}_{\alpha }{m}_{(0,0)}\otimes a_{(2,\alpha )(2,\alpha )}{▹}_{\alpha }{n}_{(0,0)}\hfill \\ \hfill =& a_{(1,\alpha )}{m}_{(-1,e)}{S}_{\alpha }\left(a_{(3,\alpha )}\right)a_{(4,\alpha )}{n}_{(-1,e)}{S}_{\alpha }\left(a_{(6,\alpha )}\right)\hfill \\ & \otimes a_{(2,\alpha )}{▹}_{\alpha }{m}_{(0,0)}\otimes a_{(5,\alpha )}{▹}_{\alpha }{n}_{(0,0)}\hfill \\ \hfill =& a_{(1,\alpha )}{m}_{(-1,e)}{n}_{(-1,e)}{S}_{\alpha }\left(a_{(4,\alpha )}\right)\otimes a_{(2,\alpha )}{▹}_{\alpha }{m}_{(0,0)}\otimes a_{(3,\alpha )}{▹}_{\alpha }{n}_{(0,0)}\hfill \\ \hfill =& a_{(1,\alpha )}{m}_{(-1,e)}{n}_{(-1,e)}{S}_{\alpha }\left(a_{(3,\alpha )}\right)\otimes a_{(2,\alpha )}{▹}_{\alpha }(m_{(0,0)}\otimes n_{(0,0)}).\hfill \end{array}$$

Thus, $M\otimes N$ is a left $\pi$-graded the Yetter–Drinfeld module over H.

Now, we prove that ${\tau }_{M,N}$ is a $\pi$-graded H-module-like map. On the one hand, it is easy to obtain:

$$\begin{array}{cc}\hfill a{▹}_{\alpha }{\tau }_{M,N}(m\otimes n)=& a{▹}_{\alpha }(\left(m_{(-1,e)}{▹}_{e}n\right)\otimes m_{(0,0)})\hfill \\ \hfill =& a_{(1,\alpha )}{▹}_{\alpha }\left(m_{(-1,e)}{▹}_{e}n\right)\otimes a_{(2,\alpha )}{▹}_{\alpha }{m}_{(0,0)}\hfill \\ \hfill =& \left(a_{(1,\alpha )}{m}_{(-1,e)}\right){▹}_{\alpha }n\otimes a_{(2,\alpha )}{▹}_{\alpha }{m}_{(0,0)}.\hfill \end{array}$$

On the other hand, we have:

$$\begin{array}{cc}& {\tau }_{M,N}\left(a{▹}_{\alpha }(m\otimes n)\right)={\tau }_{M,N}(a_{(1,\alpha )}{▹}_{\alpha }m\otimes a_{(2,\alpha )}{▹}_{\alpha }n)\hfill \\ \hfill =& {\left(a_{(1,\alpha )}{▹}_{\alpha }m\right)}_{(-1,e)}{▹}_e\left(a_{(2,\alpha )}{▹}_{\alpha }n\right)\otimes {\left(a_{(1,\alpha )}{▹}_{\alpha }m\right)}_{(0,0)}\hfill \\ \hfill =& \left(a_{(1,\alpha )(1,\alpha )}{m}_{(-1,e)}{S}_{\alpha }\left(a_{(1,\alpha )(3,\alpha )}\right)\right){▹}_e\left(a_{(2,\alpha )}{▹}_{\alpha }n\right)\otimes a_{(1,\alpha )(2,\alpha )}{▹}_{\alpha }{m}_{(0,0)}\hfill \\ \hfill =& \left(a_{(1,\alpha )}{m}_{(-1,e)}{S}_{\alpha }\left(a_{(3,\alpha )}\right)\right){▹}_e\left(a_{(4,\alpha )}{▹}_{\alpha }n\right)\otimes a_{(2,\alpha )}{▹}_{\alpha }{m}_{(0,0)}\hfill \\ \hfill =& \left(a_{(1,\alpha )}{m}_{(-1,e)}{S}_{\alpha }\left(a_{(3,\alpha )}\right)a_{(4,\alpha )}\right){▹}_{\alpha }n\otimes a_{(2,\alpha )}{▹}_{\alpha }{m}_{(0,0)}\hfill \\ \hfill =& \left(a_{(1,\alpha )}{m}_{(-1,e)}\right){▹}_{\alpha }n\otimes a_{(2,\alpha )}{▹}_{\alpha }{m}_{(0,0)}.\hfill \end{array}$$

We also check that ${\tau }_{M,N}$ is an $H_e$-comodule map as follows:

$$\begin{array}{cc}\hfill & \rho \circ {\tau }_{M,N}(m\otimes n)=\rho (\left(m_{(-1,e)}{▹}_{e}n\right)\otimes m_{(0,0)})\hfill \\ \hfill =& {\left(m_{(-1,e)}{▹}_{e}n\right)}_{(-1,e)}{m}_{(0,0)(-1,e)}\otimes {\left(m_{(-1,e)}{▹}_{e}n\right)}_{(0,0)}\otimes m_{(0,0)(0,0)}\hfill \\ \hfill =& m_{(-1,e)(1,e)}{n}_{(-1,e)}{S}_1\left(m_{(-1,e)(3,e)}\right)m_{(0,0)(-1,e)}\hfill \\ \hfill & \otimes m_{(-1,e)(2,e)}{▹}_e{n}_{(0,0)}\otimes m_{(0,0)(0,0)}\hfill \\ \hfill =& m_{(-4,1)}{n}_{(-1,e)}{S}_1\left(m_{(-2,e)}\right)m_{(-1,e)}\otimes m_{(-3,e)}{▹}_e{n}_{(0,0)}\otimes m_{(0,0)}\hfill \\ \hfill =& m_{(-2,1)}{n}_{(-1,e)}\otimes m_{(-1,e)}{▹}_e{n}_{(0,0)}\otimes m_{(0,0)}\hfill \\ \hfill =& m_{(-1,e)}{n}_{(-1,e)}\otimes \left(m_{(0,0)(-1,e)}{▹}_e{n}_{(0,0)}\right)\otimes m_{(0,0)(0,0)}\hfill \\ \hfill =& m_{(-1,e)}{n}_{(-1,e)}\otimes {\tau }_{M,N}\left(m_{(0,0)}\otimes n_{(0,0)}\right)\hfill \\ \hfill =& (i{d}_{H_1}\otimes {\tau }_{M,N})\left(m_{(-1,e)}{n}_{(-1,e)}\otimes m_{(0,0)}\otimes n_{(0,0)}\right)\hfill \\ \hfill =& (i{d}_{H_1}\otimes {\tau }_{M,N})\circ \rho (m\otimes n).\hfill \end{array}$$

For all m, $n\in M\in {}_H^{H_e}\mathcal{YD}$, define ${\tau }_{M,N}^{-1}:N\otimes M⟶$$M\otimes N$ by

$${\tau }_{M,N}^{-1}(n\otimes m)=m_{(0,0)}\otimes S_e^{-1}\left(m_{(-1,e)}\right){▹}_{e}n.$$

Then we have:

$$\begin{array}{cc}\hfill {\tau }_{M,N}\left({\tau }_{M,N}^{-1}(n\otimes m)\right)=& {\tau }_{M,N}\left(m_{(0,0)}\otimes S_1^{-1}\left(m_{(-1,e)}\right){▹}_{e}n\right)\hfill \\ \hfill =& m_{(0,0)(-1,e)}{▹}_e\left(S_1^{-1}\left(m_{(-1,e)}\right){▹}_{e}n\right)\otimes m_{(0,0)(0,0)}\hfill \\ \hfill =& m_{(-1,e)(2,e)}{▹}_e\left(S_1^{-1}\left(m_{(-1,e)(1,e)}\right){▹}_{e}n\right)\otimes m_{(0,0)}\hfill \\ \hfill =& \left(m_{(-1,e)(2,e)}{S}_1^{-1}\left(m_{(-1,e)(1,e)}\right)\right){▹}_{e}n\otimes m_{(0,0)}=n\otimes m.\hfill \end{array}$$

Similarly, ${\tau }_{M,N}^{-1}\circ {\tau }_{M,N}=i{d}_{M\otimes N}$ can be proven as follows:

$$\begin{array}{cc}& \phantom{=}{\tau }_{M,N}^{-1}\left({\tau }_{M,N}(m\otimes n)\right)\hfill \\ & ={\tau }_{M,N}^{-1}\left(\left(m_{(-1,e)}{▹}_{e}n\right)\otimes m_{(0,0)}\right)\hfill \\ & =m_{(0,0)(0,0)}\otimes S_e^{-1}\left(m_{(0,0)(-1,e)}\right){▹}_e\left(m_{(-1,e)}{▹}_{e}n\right)\hfill \\ & =m_{(0,0)(0,0)}\otimes \left(S_e^{-1}\left(m_{(0,0)(-1,e)}\right)m_{(-1,e)}\right){▹}_{e}n\hfill \\ & =m_{(0,0)}\otimes \left(S_e^{-1}\left(m_{(-1,e)}\right)m_{(-2,e)}\right){▹}_{e}n=m\otimes n,\hfill \end{array}$$

and ${\tau }_{M,N}$ is therefore a left $\pi$-graded Yetter–Drinfeld module isomorphism over H.

For any $f\in {}_H^{H_e}\mathcal{YD}(M,M^{\prime })$ and $g\in {}_H^{H_e}\mathcal{YD}(N,N^{\prime })$, we have

$$\begin{array}{cc}\hfill (g\otimes f)\circ {\tau }_{M,N}(m\otimes n)=& (g\otimes f)\left(\left(m_{(-1,e)}{▹}_{e}n\right)\otimes m_{(0,0)}\right)\hfill \\ \hfill =& g\left(m_{(-1,e)}{▹}_{e}n\right)\otimes f\left(m_{(0,0)}\right)\hfill \\ \hfill =& m_{(-1,e)}{▹}_{e}g\left(n\right)\otimes f\left(m_{(0,0)}\right)\hfill \\ \hfill =& f{\left(m\right)}_{(-1,e)}{▹}_{e}g\left(n\right)\otimes f{\left(m\right)}_{(0,0)}\hfill \\ \hfill =& {\tau }_{M^{\prime },N^{\prime }}(f\left(m\right)\otimes g\left(n\right))={\tau }_{M^{\prime },N^{\prime }}\circ (f\otimes g)(m\otimes n).\hfill \end{array}$$

This means that ${\tau }_{M,N}$ is a natural isomorphism.

In order to complete the proof, it remains to check that ${\tau }_{M,N}$ satisfies the Hexagon Axiom as follows:

$$\begin{array}{cc}\hfill & {\tau }_{M\otimes N,P}(m\otimes n\otimes p)\hfill \\ \hfill =& \left({(m\otimes n)}_{(-1,e)}{▹}_{e}p\right)\otimes {(m\otimes n)}_{(0,0)}\hfill \\ \hfill =& \left(\left(m_{(-1,e)}{n}_{(-1,e)}\right){▹}_{e}p\right)\otimes m_{(0,0)}\otimes n_{(0,0)}\hfill \\ \hfill =& \left(m_{(-1,e)}{▹}_e\left(n_{(-1,e)}{▹}_{e}p\right)\right)\otimes m_{(0,0)}\otimes n_{(0,0)}\hfill \\ \hfill =& {\tau }_{M,P}\left(m\otimes \left(n_{(-1,e)}{▹}_{e}p\right)\right)\otimes n_{(0,0)}\hfill \\ \hfill =& ({\tau }_{M,P}\otimes i{d}_N)\left(m\otimes \left(n_{(-1,e)}{▹}_{e}p\right)\otimes n_{(0,0)}\right)\hfill \\ \hfill =& ({\tau }_{M,P}\otimes i{d}_N)\left(m\otimes {\tau }_{N,P}(n\otimes p)\right)\hfill \\ \hfill =& ({\tau }_{M,P}\otimes i{d}_N)\circ (i{d}_M\otimes {\tau }_{N,P})(m\otimes n\otimes p)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\tau }_{M,N\otimes P}(m\otimes n\otimes p)\hfill \\ \hfill =& \left(m_{(-1,e)}{▹}_e(n\otimes p)\right)\otimes m_{(0,0)}\hfill \\ \hfill =& m_{(-1,1)(1,e)}{▹}_{e}n\otimes m_{(-1,e)(2,e)}{▹}_{e}p\otimes m_{(0,0)}\hfill \\ \hfill =& \left(m_{(-1,e)}{▹}_{e}n\right)\otimes \left(m_{(0,0)(-1,e)}{▹}_{e}p\right)\otimes m_{(0,0)(0,0)}\hfill \\ \hfill =& \left(m_{(-1,e)}{▹}_{e}n\right)\otimes {\tau }_{M,P}(m_{(0,0)}\otimes p)\hfill \\ \hfill =& (i{d}_N\otimes {\tau }_{M,P})(\left(m_{(-1,e)}{▹}_{e}n\right)\otimes m_{(0,0)}\otimes p)\hfill \\ \hfill =& (i{d}_N\otimes {\tau }_{M,P})({\tau }_{M,N}(m\otimes n)\otimes p)\hfill \\ \hfill =& (i{d}_N\otimes {\tau }_{M,P})\circ ({\tau }_{M,N}\otimes i{d}_P)(m\otimes n\otimes p).\hfill \end{array}$$

□

4. A Braided Hopf Algebra in ${}_{\mathbf{H}}^{{\mathbf{H}}_{\mathbf{e}}}{\mathcal{YD}}_{\mathbf{\pi }}$

Let H be a $\pi$-graded Hopf algebra. In section we will introduce the category ${}_H^{H_e}{\mathcal{YD}}_{\pi }$ different from ${}_H^H{\mathcal{YD}}_{\pi }$ and consider how to construct a braided $\pi$-graded Hopf algebra in ${}_H^{H_e}{\mathcal{YD}}_{\pi }$ and we get a generalized Radford $\pi$-graded by-product.

Definition 4.

Let H be a π-graded Hopf algebra and let A be an algebra with unit 1.

(i) A is called a left π-H-module-like algebra if A is a left π-H-module-like object with structure ${\left\{{▹}_{\alpha }\right\}}_{\alpha \in \pi }$ such that, for all $a\in H_{\lambda }$ and $x,y\in A$ with $\lambda \in \pi$,

$$\begin{array}{cc}\hfill a{▹}_{\lambda }\left(xy\right)& =\left(a_{(1,\lambda )}{▹}_{\lambda }x\right)\left(a_{(2,\lambda )}{▹}_{\lambda }y\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill a{▹}_{\lambda }1& ={\epsilon }_{\lambda }\left(a\right)1;\hfill \end{array}$$

(12)

(ii) A is said a left π-H-comodule algebra if A is endowed a family of left $H_{\alpha }$-comodule structure ${\left\{{\delta }_{\lambda }\right\}}_{\lambda \in \pi }$ such that, for all $x,y\in A$ with $\lambda \in \pi$

$$\begin{array}{cc}\hfill {\delta }_{\lambda }\left(xy\right)& =x_{(-1,\lambda )}{y}_{(-1,\lambda )}\otimes x_{(0,0)}{y}_{(0,0)},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\delta }_1\left(1\right)& =1_H\otimes 1.\hfill \end{array}$$

(14)

Definition 5.

Let $H={⨁}_{\alpha \in \pi }{H}_{\alpha }$ be a π-graded Hopf algebra. Let A be a left π-H-module-like algebra. Define a π-graded smash product $A{\#}_{\pi }H={⨁}_{\alpha \in \pi }A{\#}_{\alpha }H$ where $A{\#}_{\alpha }H:=A\otimes H_{\alpha }$, with a product $m={\{m_{\alpha ,\beta }:\left(A{\#}_{\alpha }H\right)\otimes \left(A{\#}_{\beta }H\right)⟶\left(A{\#}_{\alpha \beta }H\right)\}}_{\alpha ,\beta \in \pi }$ given by

$$\left(x{\#}_{\alpha }a\right)\left(y{\#}_{\beta }b\right)=x\left(a_{(1,\alpha )}{▹}_{\alpha }y\right){\#}_{\alpha \beta }{a}_{(2,\alpha )}b$$

for all $x,y\in A,a\in H_{\alpha },b\in H_{\beta }$ with $\alpha ,\beta \in \pi$.

The following proposition is easy to get.

Proposition 2.

With the above notation. Then $A{\#}_{\pi }H$ is a π-graded algebra if and only if A is a left π-H-module-like algebra.

Dually, we have:

Definition 6.

Let H be a π-graded Hopf algebra and let J be a co-algebra with co-unit ε.

(i) J is called a left π-H-module-like co-algebra if J is a π-A-module-like object with structure ${\left\{{▹}_{\alpha }\right\}}_{\alpha \in \pi }$ such that, for all $a\in H_{\lambda },x\in J$ with $\lambda \in \pi$

$$\begin{array}{ccc}& & \Delta \left(a{▹}_{\lambda }x\right)=\left(a_{(1,\lambda )}{▹}_{\lambda }{x}_1\right)\otimes \left(a_{(2,\lambda )}{▹}_{\lambda }{x}_2\right),\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & \epsilon \left(a{▹}_{\lambda }x\right)={\epsilon }_{\lambda }\left(a\right)\epsilon \left(x\right);\hfill \end{array}$$

(16)

(ii) J is termed a left π-H-comodule co-algebra if J is a π-H-comodule with structure ${\left\{{\delta }_{\lambda }\right\}}_{\lambda \in \pi }$ such that, for all $x\in J$ with $\alpha ,\beta \in \pi$

$$\begin{array}{ccc}& & x_{1(-1,\alpha )}{x}_{2(-1,\beta )}\otimes x_{1(0,0)}\otimes x_{2(0,0)}=x_{(-1,\alpha \beta )}\otimes x_{(0,0)1}\otimes x_{(0,0)2},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}& & x_{(-1,e)}\epsilon \left(x_{(0,0)}\right)=\epsilon \left(x\right)1_H.\hfill \end{array}$$

(18)

Definition 7.

Let $H={⨁}_{\alpha \in \pi }{H}_{\alpha }$ be a π-graded Hopf algebra. Let J be a left π-A-comodule co-algebra. Define a α-smash coproduct $J{\times }_{\alpha }H:=J\otimes H_{\alpha }$ for $\alpha \in \pi$ with following coproduct; ${\tilde{\Delta }}_{\alpha }:J{\times }_{\alpha }H⟶\left(J{\times }_{\alpha }H\right)\otimes \left(J{\times }_{\alpha }H\right)$ given by,

$${\tilde{\Delta }}_{\alpha }\left(x{\times }_{\alpha }a\right)=\left(x_1{\times }_{\alpha }{x}_{2(-1,e)}{a}_{(1,\alpha )}\right)\otimes \left(x_{2(0,0)}{\times }_{\alpha }{a}_{(2,\alpha )}\right)$$

and co-unit ${\tilde{\epsilon }}_{\alpha }:J{\times }_{\alpha }H⟶\mathbb{k}$: ${\tilde{\epsilon }}_{\alpha }\left(x{\times }_{\alpha }a\right)=\epsilon \left(x\right){\epsilon }_{\alpha }$, for all $x\in J,a\in H_{\alpha }$ and $\alpha \in \pi$.

The proof of the following proposition is straightforward.

Proposition 3.

With the above notation, then, for any $\alpha \in \pi$, $(J{\times }_{\alpha }H,{\tilde{\Delta }}_{\alpha },{\tilde{\epsilon }}_{\alpha })$ is a co-algebra if and only if J is the usual left $H_e$-comodule co-algebra.

Theorem 3.

Let Let $H={⨁}_{\alpha \in \pi }{H}_{\alpha }$ be a π-graded Hopf algebra. Let J be an algebra and a co-algebra. With the above notation. Then $J{\#}_{\pi }H$ with the co-algebra structure given as in Definition 7 (simply denoted by $J{⋉}_{\pi }H$) is a π-graded Hopf algebra without crossing and antipode if and only if the following conditions hold:

(i) J is both a left π-H-module-like algebra and a left π-H-comodule-like algebra;

(ii) J is both the usual left $H_e$-comodule co-algebra and the usual left $H_e$-comodule algebra;

(iii) ${\epsilon }_J$ is an algebra map, and $\Delta \left(xy\right)=x_1\left(x_{2(-1,e)}{▹}_e{y}_1\right)\otimes x_{2(0,0)}{y}_2$;

(iv) $a_{(1,\alpha )}{x}_{(-1,e)}\otimes a_{(2,\alpha )}{▹}_{\alpha }{x}_{(0,0)}={\left(a_{(1,\alpha )}{▹}_{\alpha }x\right)}_{(-1,e)}{a}_{(2,\alpha )}\otimes {\left(a_{(1,\alpha )}{▹}_{\alpha }x\right)}_{(0,0)}$for any $a\in H_{\alpha },x\in J$ with $\alpha \in \pi$.

Furthermore, if J has a convolution inverse $S_J\in Hom(J,J)$ of $i{d}_J$, then $J{\#}_{\pi }\times H$ is a π-graded Hopf algebra (without crossing) with antipode $\tilde{S}={\{{\tilde{S}}_{\alpha }:J{\#}_{\alpha }H⟶J{\#}_{{\alpha }^{-1}}H\}}_{\alpha \in \pi }$ described by:

$${\tilde{S}}_{\alpha }\left(x{\#}_{\alpha }a\right)=\left(1{\#}_{{\alpha }^{-1}}{S}_{\alpha }\left(x_{(-1,e)}a\right)\right)\left(S_J\left(x_{(0,0)}\right){\#}_1{1}_H\right).$$

Proof. 

The proof consists of a tedious computation similar to one in [10], left to readers. □

Theorem 4.

With the above notations. Then $J{⋉}_{\pi }H$ is a π-graded Hopf algebra if and only if J is a braided π-graded Hopf algebra in ${}_H^{H_e}\mathcal{YD}$.

Given a braided monoidal category $(\mathcal{C},\tau )$, see [1]. Recall from [11] that a braided Lie algebra $\mathcal{L}$ in $\mathcal{C}$ is an object of $\mathcal{C}$ together with a linear map $[,]:\mathcal{L}\otimes \mathcal{L}⟶\mathcal{L}$ which is a morphism in $\mathcal{C}$ satisfying:

$$\begin{array}{ccc}& \left(BLA1\right)& [,]=-[,]{\tau }_{\mathcal{L},\mathcal{L}};\hfill \\ & \left(BLA2\right)& [,](ß\otimes [,])(i+({\tau }_{\mathcal{L},\mathcal{L}}\otimes id)(ß\otimes {\tau }_{\mathcal{L},\mathcal{L}})+(ß\otimes {\tau }_{\mathcal{L},\mathcal{L}})({\tau }_{\mathcal{L},\mathcal{L}}\otimes ß))=0.\hfill \end{array}$$

Let $(J,m,\mu )$ be an algebra in $\mathcal{C}$. Then we define:

$$[,]:J\otimes J⟶J,by[,]=m-m\circ {\tau }_{J,J}.$$

If $(J,[,\left]\right)$ forms a braided Lie algebra in $\mathcal{C}$, then the resulting algebra is denoted by $J^{-}$.

Example 4.

Given any braided Lie algebra L in ${}_H^{H_e}\mathcal{YD}$. By [11], we define $U\left(L\right)=T\left(L\right)/I\left(L\right)$, where $T\left(L\right)$ is the usual tensor product algebra of L and $I\left(L\right)$ is the ideal of $T\left(L\right)$ generated by

$$\{a\otimes b-(a_{(-1,e)}▹b)\otimes a_{(0,0)}-[a,b]\}$$

for all $a,b\in L,$ and let $i_L:L⟶T\left(L\right)/I\left(L\right)$ be the canonical map, $a\mapsto \overline{a}=a+I\left(L\right)$. Then, by following the standard algebraic constructions, one has the braided universal enveloping algebra $U\left(L\right)$ in ${}_H^{H_e}\mathcal{YD}$ with the following structures: $\Delta \left(\overline{l}\right)=\overline{l}\otimes 1+1\otimes \overline{l});\Delta \left(1\right)=1\otimes 1$, $ϵ\left(\overline{l}\right)=0$; $ϵ\left(1\right)=1$, $S\left(\overline{l}\right)=-\overline{l}$ and $S\left(\overline{a}\overline{b}\right)=(S{\left(\overline{a}\right)}_{(-1,e)}▹S\left(\overline{b}\right))S{\left(\overline{a}\right)}_{(0,0)}$, for all $l\in L$ and $\overline{a}$, $\overline{b}\in U\left(L\right)$.

Furthermore, for $l_1,\cdots ,l_n\in L$, we have:

$$\Delta ({\overline{l}}_1\cdots {\overline{l}}_n)=1\otimes {\overline{l}}_1\cdots {\overline{l}}_n+\sum _{p=1}^{n-1}\sum _{\sigma }{\overline{l}}_{\sigma \left(1\right)}\cdots {\overline{l}}_{\sigma \left(p\right)}\otimes {\overline{l}}_{\sigma (p+1)}\cdots {\overline{l}}_{\sigma \left(n\right)}+{\overline{l}}_1\cdots {\overline{l}}_n\otimes 1,$$

where σ runs over all $(p,q)$-shuffles of the symmetric group $S_n$.

Then $U\left(L\right)$ is a braided Hopf algebra in ${}_H^{H_e}\mathcal{YD}$ and then by Theorem 4, we get the π-graded Hopf algebra $U\left(L\right){⋉}_{\pi }H$.

Let $V\in {}_H^{H_e}\mathcal{YD}$. We say that the braiding $\tau$ on ${}_H^{H_e}\mathcal{YD}$ is symmetric on V if ${\tau }_{V,V}^2=i{d}_{V\otimes V}$. That means, for $v,w\in V$

$$v_{(-1,e)}▹w\otimes v_{(0,0)}=w_{(0,0)}\otimes S_1^{-1}\left(w_{(-1,e)}\right)▹v.$$

Theorem 5.

Let H be a π-graded Hopf algebra and J an algebra in ${}_H^{H_e}\mathcal{YD}$. Assume that the braiding ${\tau }_{J,J}$ for J is symmetric. Then J is a derived Lie algebra induced by the algebra J in ${}_H^{H_e}\mathcal{YD}$, which is defined by the Lie product

$$[,]:J\otimes J⟶Jby[a,b]=ab-\sum (a_{(-1,e)}▹b)a_{(0,0)}$$

for all $a\in J$.

Proof. 

Straightforward. □

5. Group Double Centralizer Properties

In this section we introduce the notion of a $\pi$-double centralizer property and then obtain a $\pi$-double centralizer property. By the way, we remark here that the (semi)group grading can be viewed as a Hopf algebra co-action. Double constructions (grading plus Hopf algebra coaction) are important like a double complex and work in quantum mechanics and other mainstream subjects (see [12,13]).

Let A be a $\pi$-graded algebra and B an algebra. Assume that M is a left $\pi$-A-module-like object with a $\pi$-action ${\left\{{▹}_{\alpha }\right\}}_{\alpha \in \pi }$ and a right B-module such that $\left(a{▹}_{\alpha }m\right)·x=a{▹}_{\alpha }(m·x)$ for $a\in A_{\alpha }$ and $x\in B$ with $\alpha \in \pi$. Then $\varphi$: $A⟶En{d}_B\left(M\right),\varphi \left(a\right)\left(m\right)=a{▹}_{\alpha }m$ and $\psi$: $B⟶En{d}_A\left(M\right),\psi \left(x\right)\left(m\right)=m·x$ are two natural $\pi$-algebra homomorphisms.

Definition 8.

If both ϕ and ψ are surjective, we say that the $(A,B,M)$ satisfies π-double centralizer property.

Remark 1.

(1) When π is trivial, then π-double centralizer property is the classical Schur–Weyl duality;

(2) If M is a left π-A-module-like object with structure ${\left\{{▹}_{\alpha }\right\}}_{\alpha \in \pi }$, then we say that the map ϕ: $A⟶End\left(M\right)$, $\varphi \left(a\right)\left(m\right)=a{▹}_{\alpha }m$ is the corresponding π-representation.

Let A be a $\pi$-graded Hopf algebra and let V be a n-dimensional vector space in ${}_A^{A_e}\mathcal{YD}$. Then V is a left $\pi$-A-module-like object. Let $\Phi$: $A⟶End\left(V\right)$ be the corresponding $\pi$-representation. Then A$\pi$-acts diagonally on $V^{\otimes n}$, and we have the resulting $\pi$-representation ${\Phi }^n={\Phi }^{\otimes n}\circ {\Delta }_{n-1}:A⟶End{V}^{\otimes n}$ such that for all $v_1,\cdots ,v_n\in V$, the following equality holds

$${\Phi }^n\left(a\right)(v_1\otimes \cdots \otimes v_n)=a_{(1,\alpha )}{▹}_{\alpha }{v}_1\otimes \cdots \otimes a_{(n,\alpha )}{▹}_{\alpha }{v}_n$$

(19)

for $a\in A_{\alpha }$ with $\alpha \in \pi$.

One writes $S_n$ in the form of generators $\{{\sigma }_i=(i,i+1)\mid i=1,\cdots ,n-1\}$. Then, we have an action of $S_n$ on $V^{\otimes n}$ by:

$$(i,i+1)·(v_1\otimes \cdots \otimes v_n)=v_1\otimes \cdots \otimes v_{i(-1,e)}▹v_{i+1}\otimes v_{i(0,0)}\otimes \cdots \otimes v_n.$$

Then, extending the above action multiplicatively to $S_n$ and also linearly to the Hopf algebra $\mathbb{k}{S}_n$ so that we can obtain the following representation:

$$\Psi :\mathbb{k}{S}_n⟶End{V}^{\otimes n}.$$

(20)

We can now see that the action of $\mathbb{k}{S}_n$ on $V^{\otimes n}$ induces an action of $\mathbb{k}{S}_n$ on ${\left(EndV\right)}^{\otimes n}$, that is, for all $f_i\in EndV$ and $v_j\in V$, we have:

$$\begin{array}{cc}\hfill & ((i,i+1)·(f_1\otimes \cdots \otimes f_n))(v_1\otimes \cdots \otimes v_n)\hfill \\ \hfill =& (i,i+1)·(f_1\otimes \cdots \otimes f_n)((i,i+1)·(v_1\otimes \cdots \otimes v_n))\hfill \\ \hfill =& (i,i+1)·\left((f_1\otimes \cdots \otimes f_n)(v_1\otimes \cdots \otimes v_{i(-1,e)}▹v_{i+1}\otimes v_{i(0,0)}\otimes \cdots \otimes v_n)\right)\hfill \\ \hfill =& (f_1\left(v_1\right)\otimes \cdots \otimes {\left(f_i(v_{i(-1,e)}▹v_{i+1})\right)}_{(-1,e)}▹f_{i+1}\left(v_{i(0,0)}\right)\hfill \\ \hfill & \otimes {\left(f_i(v_{i(-1,e)}▹v_{i+1})\right)}_{(0,0)}\otimes \cdots \otimes f_n\left(v_n\right).\hfill \\ \hfill =& (f_1\left(v_1\right)\otimes \cdots \otimes {\left(f_i{\left(v_{i+1}\right)}_{(0,0)}\right)}_{(-1,e)}▹f_{i+1}(S_1^{-1}\left({\left(v_{i+1}\right)}_{(-1,e)}\right)▹v_i)\hfill \\ \hfill & \otimes {\left(f_i\left({\left(v_{i+1}\right)}_{(0,0)}\right)\right)}_{(0,0)}\otimes \cdots \otimes f_n\left(v_n\right))\hfill \\ \hfill =& (f_1\left(v_1\right)\otimes \cdots \otimes f_{i(-1,e)}·{\left({\left(v_{i+1}\right)}_{(0,0)}\right)}_{(-1,e)}▹f_{i+1}(S_1^{-1}\left({\left(v_{i+1}\right)}_{(-1,e)}\right)▹v_i)\hfill \\ \hfill & \otimes f_{i(0,0)}\left({\left(v_{i+1}\right)}_{(0,0)(0,0)}\right)\otimes \cdots \otimes f_n\left(v_n\right)),\hfill \end{array}$$

since $f_i\in EndV$.

By putting $f_{i+1}=id$, we have:

$$\begin{array}{cc}\hfill & ((i,i+1)·(f_1\otimes \cdots \otimes f_n))(v_1\otimes \cdots \otimes v_n)\hfill \\ \hfill =& (f_1\left(v_1\right)\otimes \cdots \otimes f_{i(-1,1)}▹{\left({\left(v_{i+1}\right)}_{(-1,e)}\right)}_2▹S_1^{-1}{\left({\left(v_{i+1}\right)}_{(-1)}\right)}_1)▹v_i)\hfill \\ \hfill & \otimes f_{i\left(0\right)}\left({\left(v_{i+1}\right)}_0\right)\otimes \cdots \otimes f_n\left(v_n\right))\hfill \\ \hfill =& (f_1\left(v_1\right)\otimes \cdots \otimes f_{i(-1,e)}▹v_i)\otimes f_{i(0,0)}\left(v_{i+1}\right)\otimes \cdots \otimes f_n\left(v_n\right)).\hfill \end{array}$$

Thus, we obtain the following formula:

$$\begin{array}{c}\hfill (i,i+1)·(f_1\otimes \cdots \otimes f_i\otimes id\otimes f_{i+2}\otimes \cdots \otimes f_n)\\ \hfill =f_1\otimes \cdots \otimes f_{i(-1,e)}\otimes f_{i(0,0)}\otimes f_{i+2}\otimes \cdots \otimes f_n.\end{array}$$

(21)

Proposition 4.

The π-representation ${\Phi }^n$ of A and the representation Ψ of $\mathbb{k}{S}_n$, as described in (19) and (20), commutes each other in $End{V}^{\otimes n}$.

Proof. 

Straightforward. □

Proposition 5.

Let V be an J-module in ${}_A^{A_e}\mathcal{YD}$ and J a co-commutative bi-algebra in ${}_A^{A_e}\mathcal{YD}$. Suppose that the map ${\tau }_{V,V}$ for V is symmetric and ${\tau }_{A,A}$ for J is also symmetric. Then the actions of J and $\mathbb{k}{S}_n$ on $V^{\otimes n}$ commutes each other.

Proof. 

We only need to prove the proposition for the case of $n=2$ and $\sigma ={\tau }_{J,J}$. For $v\otimes w\in V\otimes V$, and $x\in J$. If $\varphi$: $J\otimes V⟶V$ denotes the action of J on V, then the action of x on $v\otimes w$ is given by $a·(v\otimes w)=x_1·({\left(x_2\right)}_{(-1,e)}▹v)\otimes {\left(a_2\right)}_{(0,0)}·w$, where $\Delta \left(x\right)=x_1\otimes x_2$. We prove that $(x\circ {\tau }_{J,J})(v\otimes w)=({\tau }_{J,J}\circ x)(v\otimes w)$. Indeed,

$$\begin{array}{cc}\hfill & (x\circ {\tau }_{J,J})(v\otimes w)=x·(v_{(-1,e)}▹w\otimes v_{(0,0)})\hfill \\ \hfill =& x_1·({\left(x_2\right)}_{(-1,e)}{v}_{(-1,e)}▹w)\otimes {\left(x_2\right)}_{(0,0)}·v_{(0,0)}\hfill \\ \hfill =& x_1·({(x_2·v)}_{(-1,e)}▹w)\otimes {(x_2·v)}_{(0,0)}\hfill \\ \hfill & (\mathrm{by} \mathrm{assumption}{\rho }_V(x·v)=x_{(-1,e)}{v}_{(-1,e)}\otimes x_{(0,0)}·v_{(0,0)})\hfill \\ \hfill =& x_1·w_0\otimes S_1^{-1}\left(w_{(-1,e)}\right)▹(x_2·v).\hfill \end{array}$$

On the other hand, we have that:

$$\begin{array}{cc}\hfill & {\tau }_{J,J}(x·(v\otimes w))={\tau }_{{\rho }_A}(x_1·({\left(x_2\right)}_{(-1,e)}▹v)\otimes {\left(x_2\right)}_{(0,0)}·w)\hfill \\ \hfill =& {(x_1·({\left(x_2\right)}_{(-1,e)}·v))}_{(-1,e)}▹{\left(x_2\right)}_{(0,0)}·w\otimes {(x_1·({\left(x_2\right)}_{(-1,e)}▹v))}_{(0,0)}\hfill \\ \hfill =& {({\left(x_2\right)}_0·w)}_0\otimes S_1^{-1}\left({({\left(x_2\right)}_{(0,0)}·w)}_{(-1,e)}\right)▹(x_1·({\left(x_2\right)}_{(-1,e)}▹v))\hfill \\ \hfill =& {\left(x_2\right)}_{(0,0)(0,0)}·w_{(0,0)}\otimes S_1^{-1}\left({\left({\left(x_2\right)}_{(0,0)}\right)}_{(-1,e)}{w}_{(-1,e)}\right)▹(x_1·({\left(x_2\right)}_{(-1,e)}▹v))\hfill \\ \hfill =& {\left(x_2\right)}_{(0,0)}·w_{(0,0)}\otimes S_1^{-1}\left({\left({\left(x_2\right)}_{(-1,e)}\right)}_2{w}_{(-1,e)}\right)▹(x_1·({\left({\left(x_2\right)}_{(-1,e)}\right)}_1·v)\hfill \\ \hfill =& {\left(x_2\right)}_{(0,0)}·w_{(0,0)}\otimes S_1^{-1}\left(w_{(-1,e)}\right)S_1^{-1}\left({\left({\left(x_2\right)}_{(-1,e)}\right)}_2\right)·(x_1·\left({\left({\left(x_2\right)}_{(-1,e)}\right)}_1\right)·v)\hfill \\ \hfill =& {\left(x_2\right)}_{(0,0)}·w_{(0,0)}\otimes S_1^{-1}\left(w_{(-1,e)}\right)▹((S_1^{-1}\left({\left({\left(x_2\right)}_{(-1,e)}\right)}_3\right)·A_e)\hfill \\ \hfill & ·(\left(S_1^{-1}\left({\left({\left(x_2\right)}_{(-1,e)}\right)}_2\right){\left({\left(x_2\right)}_{(-1,e)}\right)}_1\right)·v)\left)\right)\hfill \\ \hfill =& {\left(x_2\right)}_{(0,0)}·w_{(0,0)}\otimes S_1^{-1}\left(w_{(-1,e)}\right)(S_1^{-1}\left({\left(x_2\right)}_{(-1,e)}\right)▹x_1)·v\hfill \\ \hfill =& ({\left(x_1\right)}_{(-1,e)}▹x_2)·w_{(0,0)}\otimes S_1^{-1}\left(w_{(-1,e)}\right)({\left(x_1\right)}_0·v)\hfill \\ \hfill =& x_1·w_{(0,0)}\otimes S_1^{-1}\left(w_{(-1,e)}\right)▹(x_2·v),\hfill \end{array}$$

since ${\tau }_{J,J}\Delta \left(x\right)=\Delta \left(x\right)$, for all $x\in J$ by the assumption, $(x_1\otimes x_2={\left(x_1\right)}_{(-1,e)}▹x_2\otimes {\left(x_1\right)}_{(0,0)})$.

This completes the proof. □

Let V be a n-dimensional vector space in ${}_A^{A_e}\mathcal{YD}$. Then $EndV$ is an associative algebra in ${}_A^{A_e}\mathcal{YD}$. By Theorem 3, we have the braided Lie algebra $g{l}_n\left(V\right):=End{\left(V\right)}^{-}$ in ${}_A^{A_e}\mathcal{YD}$ if ${\tau }_{V,V}$ is symmetric on V. By Example 4, we can construct the braided universal enveloping algebra $U\left(g{l}_n\left(V\right)\right)$ of $g{l}_n\left(V\right)$ which is not a Hopf algebra. Then by Theorem 2, we have:

Proposition 6.

Let A be a π-graded Hopf algebra with bijective antipode $S_A$. Let $V\in {}_A^{A_e}\mathcal{YD}$ be a n-dimensional vector space. Assume that the natural twisting map ${\tau }_{EndV,EndV}$ for $EndV$ is symmetric. Then the by-product $K=U\left(g{l}_n\left(V\right)\right){⋉}_{\pi }A$ is a π-graded Hopf algebra (without crossing) with the multiplication, the coproduct and the antipode given by:

$$\begin{array}{ccc}& & \left(x{\#}_{\alpha }a\right)\left(y{\#}_{\beta }b\right)=x(a_{(1,\alpha )}▹y){\#}_{\alpha \beta }{a}_{(2,\alpha )}b,\hfill \\ & & {\tilde{\Delta }}_{\alpha }\left(x{\#}_{\alpha }a\right)=\left(x_1{\#}_{\alpha }{x}_{2(-1,1)}{a}_{(1,\alpha )}\right)\otimes \left(x_{2(0,0)}{\#}_{\alpha }{a}_{(2,\alpha )}\right)\hfill \\ & & {\tilde{S}}_{\alpha }\left(x{\#}_{\alpha }a\right)=\left(1{\#}_{{\alpha }^{-1}}{S}_{\alpha }\left(x_{(-1,e)}a\right)\right)\left(S_{U\left(g{l}_n\left(V\right)\right)}\left(x_{(0,0)}\right){\#}_e{1}_H\right)\hfill \end{array}$$

for all $x,y\in U\left(g{l}_n\left(V\right)\right),a\in A_{\alpha },b\in A_{\beta }$ with $\alpha ,\beta \in \pi$.

By Propositions 3 and 6 and Example 4, we have the following proposition:

Proposition 7.

Let J be a co-commutative Hopf algebra in ${}_A^{A_e}\mathcal{YD}$ and let V be a n-dimensional J-module in ${}_A^{A_e}\mathcal{YD}$ such that ${\tau }_{V,V}$ is symmetric. Suppose that the map ${\tau }_{J,J}$ for J is also symmetric. Then the π-action of $J{⋉}_{\pi }A$ and the action of $\mathbb{k}{S}_n$ on $V^{\otimes n}$ commute. In particular, if the braiding map for $EndV$ is symmetric, then the π-action of $K=U\left(g{l}_n\left(V\right)\right){⋉}_{\pi }A$ and the action of $\mathbb{k}{S}_n$ commutes each other in $End{V}^{\otimes n}$.

Proof. 

The proof follows from Propositions 2–4 and Theorem 3. □

Lemma 1.

Given the π-graded by-product $K=U\left(g{l}_n\left(V\right)\right){⋉}_{\pi }A$ and the representation Ψ: $\mathbb{k}{S}_n⟶End{V}^{\otimes n}$ as given in (20) and the π-representation ${\Phi }^n:K⟶End{V}^{\otimes n}$, where ${\Phi }^n={\Phi }^{\left(n\right)}·{\Delta }_{n-1}$. Then, the following formulae,

(a) 

${\Delta }_n(f\#1)={\sum }_{j=0}^n(1\#f_{(-j,e)})\otimes \cdots \otimes (1\#f_{(-1,e)})\otimes (f_{(0,0)}\#1)\otimes 1^{\otimes n-j}$ and

(b) 

${\Phi }^{n+1}(f\#1)=(1+(2,1)+\cdots +(n+1,n)\cdots (2,1))\circ (f\otimes 1^{\otimes n})$

hold, for all $f\in EndV$.

Proof. 

The (a) and (b) can be proved by induction on n. Left to readers. □

Theorem 6.

Let A be a π-graded Hopf algebra with bijective antipode $S_A$. Let V be a n-dimensional vector space in ${}_A^{A_e}\mathcal{YD}$. Assume that ${\tau }_{V,V}$ and are symmetric. Construct the π-graded Hopf algebra $K=U\left(g{l}_n\left(V\right)\right){⋉}_{\pi }A$. Let $\Phi :K⟶EndV$ be the corresponding π-representation as before. If $n!\in {\mathbb{k}}^{\times }$, then $(K,\mathbb{k}{S}_n,V^{\otimes n})$ has π-double centralizer property.

Proof. 

According to Definition 8, we want to show: (i) $En{d}_{\Psi \left(\mathbb{k}{S}_n\right)}{V}^{\otimes n}={\Phi }^n\left(K\right)$, and (ii) $En{d}_{{\Phi }^n\left(K\right)}{V}^{\otimes n}=\Psi \left(\mathbb{k}{S}_n\right)$. We here only check (i). Part (ii) can be checked similarly.

Obviously, ${\Phi }^n\left(K\right)\subset \Phi {\left(K\right)}^{\otimes n}$ by Lemma 1. Since the images ${\Phi }^n\left(K\right)$ and $\Psi \left(\mathbb{k}{S}_n\right)$ commutes each other in $End{V}^{\otimes n}$ by Proposition 7, we have ${\Phi }^n\left(K\right)\subseteq En{d}_{\Psi \left(\mathbb{k}{S}_n\right)}{V}^{\otimes n}$ under the adjoint action of $\mathbb{k}{S}_n$ on $\Phi {\left(K\right)}^{\otimes n}$ by (21). In order to finishing the proof of the part (a), one now just proves another inclusion of equation (a). We will use induction on n.

Denote by $t={\sum }_{\sigma \in S_n}\sigma$ and let the action of t on $\Phi {\left(K\right)}^{\otimes n}$ be an adjoint action described by (13). Then, for $n=1$, the assertion trivially holds. Let $a=1+(2,1)+\cdots +(n,n-1)\cdots (2,1)\in \mathbb{k}{S}_n$ and $t^{\prime }={\sum }_{\sigma \in S_{\{2,\cdots ,n\}}}\sigma$. Then, we have $t=a{t}^{\prime }$, and thus, for all $f_1\otimes \cdots \otimes f_n\in En{d}_{\Psi \left(\mathbb{k}{S}_n\right)}{V}^{\otimes n}$, for all $f_i\in \Phi \left(K\right)$, we have $t·(f_1\otimes \cdots \otimes f_n)=a{t}^{\prime }·(f_1\otimes \cdots \otimes f_n)=a·(f_1\otimes t^{\prime }·(f_1\otimes \cdots f_n))$. But $t^{\prime }·(f_2\otimes \cdots \otimes f_n)\in En{d}_{\Psi \left(\mathbb{k}{S}_{\{2,\cdots ,n\}}\right)}{V}^{\otimes n-1}$, hence by the induction hypothesis, we have: $t^{\prime }·(f_2\otimes \cdots \otimes f_n)\in {\Phi }^{n-1}\left(K\right)$. This shows that there exists a $g=\sum b_i\#h_i\in K$ with $t^{\prime }·(f_2\otimes \cdots \otimes f_n)={\Phi }^{n-1}\left(g\right)$. If ${\Delta }^{n-2}\left(g\right)=g_{\left(1\right)}\otimes \cdots \otimes g_{(n-1)}$, then we deduce that:

$$\begin{array}{cc}\hfill t·(f_1\otimes \cdots \otimes f_n)& =a·f_1\otimes {\Phi }^{\otimes n-1}(g_{\left(1\right)}\otimes \cdots \otimes g_{(n-1)})\hfill \\ \hfill & =a·f_1{\Phi }^{\otimes n}\left(S\left(g_{\left(1\right)}\right)g_{\left(2\right)}\right)\otimes g_{\left(3\right)}\otimes \cdots \otimes g_{(n+1)})\hfill \\ \hfill & =a·(f_1\Phi \left(S{g}_{\left(1\right)}\right)\otimes 1^{\otimes n-1})\left({\Phi }^{\otimes n}(g_{\left(2\right)}\otimes g_{\left(3\right)}\otimes \cdots \otimes g_{(n+1)})\right)\hfill \\ \hfill & =a·\left((f_1\Phi \left(S\left(g_{\left(1\right)}\right)\right)\otimes 1^{\otimes n-1}){\Phi }^n\left(g_{\left(2\right)}\right)\right).\hfill \end{array}$$

Since we see that ${\Phi }^n\left(g_{\left(2\right)}\right)$ is fixed by $\mathbb{k}{S}_n$, hence, by Lemma 1 the above equation is equal to:

$$a·(f_1\Phi \left(S\left(g_{\left(1\right)}\right)\right)\otimes 1^{\otimes n-1}){\Phi }^n\left(g_{\left(2\right)}\right))={\Phi }^n(f_1\Phi S{g}_{\left(1\right)}\#1){\Phi }^n\left(g_{\left(2\right)}\right),$$

which is clearly in ${\Phi }^n\left(K\right)$. Hence $f_1\otimes \cdots \otimes f_n\in {\Phi }^n\left(K\right)$, by our assumption that $n!\in {\mathbb{k}}^{\times }$. □

6. Examples and Special Cases

In this section we will treat some examples and special cases. The main purpose is to illustrate results in the previous sections about $\pi$-graded Yetter–Drinfeld modules.

6.1. A Long $\pi$-Dimodule Category ${}_B^H{\mathcal{L}}_{\pi }$

The following is a generalization of the one in [14].

Definition 9.

Let B be a Hopf algebra and let H be a π-graded Hopf algebra. A left B-H-Long α-dimodule M is both a left B-module with a structure ▹ and a left $H_{\alpha }$-comodule with structure ${\rho }_{\alpha }^M$ with $\alpha \in \pi$, such that:

$${\rho }_{\alpha }^M(a▹m)=m_{(-1,\alpha )}\otimes a▹m_{(0,0)},$$

(22)

where we write ${\rho }_{\alpha }^M\left(m\right)=m_{(-1,\alpha )}\otimes m_{(0,0)}$, for any $a\in B$ and $m\in M$.

Now, we denote by ${}_B^H{\mathcal{L}}_{\alpha }$ a category of left B-H-Long $\alpha$-dimodules and we can form the category ${}_B^H{\mathcal{L}}_{\pi }={⨁}_{\alpha \in \pi }{}_B^H{\mathcal{L}}_{\alpha }$ which the composition of morphism of B-H-Long $\alpha$-dimodules is the standard composition of the underlying linear maps.

The proof of the following lemma is straightforward.

Lemma 2.

The category ${}_B^H\mathcal{L}$ is a monoidal category. Moreover, for any $\alpha ,\beta \in \pi$, let $M\in {}_B^H{\mathcal{L}}_{\alpha }$ and $N\in {}_B^H{\mathcal{L}}_{\beta }$. Then $M\otimes N\in {}_B^H{\mathcal{L}}_{\alpha \beta }$ with the following structures:

$$\begin{array}{ccc}& & a▹(m\otimes n)=\sum A_e▹m\otimes a_2▹n\hfill \end{array}$$

(23)

$$\begin{array}{ccc}& & {\rho }_{\alpha \beta }^{M\otimes N}(m\otimes n)=m_{(-1,\alpha )}{n}_{(-1,\beta )}\otimes m_{(0,0)}\otimes n_{(0,0)}\hfill \end{array}$$

(24)

for any $\alpha ,\beta \in \pi ,m\in M$ and $n\in N$.

Recall that a Hopf algebra H is called quasitriangular if there exists an invertible element $R=R^{\left(1\right)}\otimes$$R^{\left(2\right)}\in H\otimes H$ satisfying the following conditions:

$$\begin{array}{cc}\left(TQT1\right)\hfill & \left(\Delta \otimes \iota \right)\left(R\right)=R^{\left(1\right)}\otimes r^{\left(1\right)}\otimes R^{\left(2\right)}{r}^{\left(2\right)},\hfill \\ \left(TQT2\right)\hfill & \left(\iota \otimes \Delta \right)\left(R\right)=R^{\left(1\right)}{r}^{\left(1\right)}\otimes r^{\left(2\right)}\otimes R^{\left(2\right)},\hfill \\ \left(TQT3\right)\hfill & R{\Delta }_{\alpha }\left(h\right)={\Delta }^{cop}\left(h\right)R.\hfill \end{array}$$

Furthermore, we say that a quasitriangular Hopf algebra $(H,R)$ is triangular if $R^{-1}=R^{\left(2\right)}\otimes R^{\left(1\right)}$.

As a direct result of the above definition, we have:

$$\begin{array}{c}\epsilon \left(R^{\left(1\right)}\right)R^{\left(2\right)}=R^{\left(1\right)}\epsilon \left(R^{\left(2\right)}\right)=1,\\ R^{-1}=R^{\left(1\right)}\otimes S^{-1}\left(R^{\left(2\right)}\right)=S\left(R^{\left(1\right)}\right)\otimes R^{\left(2\right)}.\end{array}$$

Definition 10.

Let H be a crossed π-graded Hopf algebra. We call H coquasitriangular if it is endowed with a family $\sigma ={\left\{{\sigma }_{\alpha ,\beta }:H_{\alpha }\otimes H_{\beta }⟶\mathbb{k}\right\}}_{\alpha ,\beta \in \pi }$ of $\mathbb{k}$-linear maps such that ${\sigma }_{\alpha ,\beta }$ is invertible for any $\alpha ,\beta \in \pi$ and the following conditions are satisfied.

$$\begin{array}{cc}\left(TCT1\right)\hfill & {\sigma }_{\alpha \beta ,\gamma }(xy,z)={\sigma }_{\alpha ,\gamma }\left(x,z_{(2,\gamma )}\right){\sigma }_{\beta ,\gamma }\left(y,z_{(1,\gamma )}\right),\hfill \\ \left(TCT2\right)\hfill & {\sigma }_{\alpha ,\beta \gamma }(x,yz)={\sigma }_{\alpha ,\beta }\left(x_{(1,\alpha )},y\right){\sigma }_{{\beta }^{-1}\alpha \beta ,\gamma }\left({\psi }_{{\beta }^{-1}}\left(x_{(2,\alpha )}\right),z\right),\hfill \\ \left(TCT3\right)\hfill & {\sigma }_{\alpha ,\beta }\left(x_{(1,\alpha )},y_{(1,\beta )}\right)y_{(2,\beta )}{\psi }_{{\beta }^{-1}}\left(x_{(2,\alpha )}\right)=x_{(1,\alpha )}{y}_{(1,\beta )}{\sigma }_{\alpha ,\beta }\left(x_{(2,\alpha )},y_{(2,\beta )}\right),\hfill \\ \left(TCT4\right)\hfill & {\sigma }_{\alpha ,\gamma }(x,y)={\sigma }_{\beta \alpha {\beta }^{-1},\beta \gamma {\beta }^{-1}}\left({\psi }_{\beta }\left(x\right),{\psi }_{\beta }\left(y\right)\right),\hfill \end{array}$$

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

Furthermore, we say that H is π-cotriangular if:

$${\sigma }_{\alpha ,\beta }^{-1}(a,b)={\sigma }_{\beta ,\alpha }\left({\psi }_{\beta }\left(b\right),a\right)$$

for all $a\in B_{\alpha }$ and $b\in B_{\beta }$ with $\alpha ,\beta \in \pi$.

Remark 2.

(1) We note that $\left(H_e,{\sigma }_{1,1}\right)$ is the usual coquasitriangular Hopf algebra;

(2) As a direct result of the above definition, we have

$$\begin{array}{c}{\sigma }_{1,\alpha }\left(1,a\right)={\epsilon }_{\alpha }\left(a\right)1={\sigma }_{\alpha ,1}\left(a,1\right)fora\in B_{\alpha },\\ {\sigma }_{\alpha ,\beta }^{-1}(a,b)={\sigma }_{{\alpha }^{-1},\beta }\left(S_{{\alpha }^{-1}}^{-1}\left(a\right),b\right)={\sigma }_{{\beta }^{-1}\alpha \beta ,{\beta }^{-1}}\left({\psi }_{{\beta }^{-1}}\left(a\right),S_{\beta }\left(b\right)\right)fora\in B_{\alpha },b\in B_{\beta };\end{array}$$

(3) When π is trivial, we have the notion of the usual coquasitriangular Hopf algebra.

Example 5.

If $\left(H,\sigma \right)$ is coquasitriangular then any $M\in {}^{H_{\alpha }}\mathcal{M}$ can be regarded as a left π-H-module-like object with π-action:

$$h{▹}_{\lambda }m={\sigma }_{\lambda ,\alpha }(h,m_{(-1,\alpha )})m_{(0,0)},forallh\in H_{\lambda },m\in M.$$

In fact, we have: $1{▹}_{e}m={\sigma }_{1,\alpha }(1,m_{(-1,\alpha )})m_{(0,0)}={\epsilon }_{\alpha }\left(m_{(-1,\alpha )}\right)m_{(0,0)}=m$ and for any $h\in H_{\lambda },g\in H_{\beta }$ and $m\in M$,

$$\begin{array}{cc}\hfill g{▹}_{\beta }\left(h{▹}_{\lambda }m\right)=& {\sigma }_{\lambda ,\alpha }(h,m_{(-1,\alpha )})g{▹}_{\beta }{m}_{(0,0)}\hfill \\ \hfill =& {\sigma }_{\lambda ,\alpha }(h,m_{(-1,\alpha )}){\sigma }_{\beta ,\alpha }(g,m_{(0,0)(-1,\alpha )})m_{(0,0)(0,0)}\hfill \\ \hfill =& {\sigma }_{\lambda ,\alpha }(h,m_{(-1,\alpha )(1,\alpha )}){\sigma }_{\beta ,\alpha }(g,m_{(-1,\alpha )(2,\alpha )})m_{(0,0)}\hfill \\ \hfill \stackrel{\left(TCT1\right)}{=}& {\sigma }_{\beta \lambda ,\alpha }(gh,m_{(-1,\alpha )})m_{(0,0)}=\left(gh\right){▹}_{\beta \lambda }m.\hfill \end{array}$$

Let H be a Hopf algebra and let $B={⨁}_{\alpha \in \pi }{B}_{\alpha }$ be a crossed $\pi$-graded algebra. Then $H\otimes B_e$ be a tensor Hopf algebra and $H\otimes B={⨁}_{\alpha \in \pi }(H\otimes B_{\alpha })$ be a crossed $\pi$-graded Hopf algebra. We have the main result of this section as follows.

Theorem 7.

Let $(H,R)$ be a quasitriangular Hopf algebra and $(B,\sigma )$ coquasitriangular π-graded Hopf algebra with crossing ψ. Then the category ${}_H^B{\mathcal{L}}_{\pi }$ is a braided monoidal subcategory of the π-graded Yetter–Drinfeld category ${}_{H\otimes B}^{H\otimes B_e}\mathcal{YD}$ under the following π-action and comodule coaction given by:

$$\left\{\begin{array}{c}(a\otimes x){▹}_{\alpha }m={\sigma }_{\alpha ,\beta }(x,m_{(-1,\beta )})a·m_{(0,0)},\hfill \\ {\rho }_M\left(m\right):=m_{[-1,e]}\otimes m_{\left[0\right]}:=R^{\left(2\right)}\otimes m_{(-1,e)}\otimes R^{\left(1\right)}·m_{(0,0)},\hfill \end{array}\right.$$

for $a\in H$, $x\in B_{\alpha }$ and $m\in M\in {}_H^B{\mathcal{L}}_{\pi }$. The braiding on ${}_H^B{\mathcal{L}}_{\pi }$ is given by:

$${\tau }_{M,N}:M\otimes N⟶N\otimes M;$$

$$m\otimes n⟼{\sigma }_{\alpha {\beta }^{-1}{\alpha }^{-1},\alpha }({\psi }_{\alpha }{S}_{\beta }^{-1}\left(n_{(-1,\beta )}\right),m_{(-1,\alpha )})R^{\left(1\right)}·n_{(0,0)}\otimes S_1^{-1}\left(R^{\left(2\right)}\right)·m_{(0,0)}$$

(25)

for all $M,N\in {}_H^B\mathcal{L}$, $m\in M$ and $n\in N$.

Furthermore, if $(H,R)$ is triangular and $(B,\sigma )$ is π-cotriangular, then ${}_H^B{\mathcal{L}}_{\pi }$ is symmetric.

Proof. 

We first check that M is a left $\pi$-$H\otimes B$-module-like object. Indeed, for all $a,b\in H$, $x\in B_{\alpha },y{\in }_{\beta },m\in M\in {}_H^B{\mathcal{L}}_{\gamma }$ and $\alpha ,\beta ,\gamma \in \pi$

$$\begin{array}{cc}\hfill & \left((a\otimes x)(b\otimes y)\right){▹}_{\alpha \beta }m={\sigma }_{\alpha \beta ,\gamma }(xy,m_{(-1,\gamma )})\left(ab\right)·m_{(0,0)}\hfill \\ \hfill \stackrel{\left(TCT1\right)}{=}& {\sigma }_{\alpha ,\gamma }(x,m_{(-1,\gamma )(2,\gamma )}){\sigma }_{\beta ,\gamma }(y,m_{(-1,\gamma )(1,\gamma )})(\left(ab\right)·m_{(0,0)})\hfill \\ \hfill =& {\sigma }_{\alpha ,\gamma }(x,m_{(0,0)(-1,\gamma )}){\sigma }_{\beta ,\gamma }(y,m_{(-1,\gamma )})(a·(b·m_{(0,0)(0,0)}))\hfill \\ \hfill \stackrel{\left(11\right)}{=}& {\sigma }_{\alpha ,\gamma }(x,{(b·m_{(0,0)})}_{(-1,\gamma )}){\sigma }_{\beta ,\gamma }(y,m_{(-1,\gamma )})(a·{(b·m_{(0,0)})}_{(0,0)}))\hfill \\ \hfill =& {\sigma }_{\beta ,\gamma }(y,m_{(-1,\gamma )})\left((a\otimes x){▹}_{\alpha }(b·m_{(0,0)})\right)\hfill \\ \hfill =& (a\otimes x){▹}_{\alpha }\left((b\otimes y){▹}_{\beta }m\right).\hfill \end{array}$$

Next, for any $\alpha \in \pi$, by (11) and (TQT2), one can gets that that the equation $({\Delta }_{\alpha }\otimes {\iota }_M)\circ {\rho }_{\alpha }^M=({\iota }_{{(H\otimes B)}_{\alpha }}\otimes {\rho }_{\alpha }^M)\circ {\rho }_{\alpha }^M$ holds, i.e, M is a left ${(H\otimes B)}_{\alpha }$-comodule.

Then we show (7) holds. For $a\in H,x\in B_{\alpha }$ and $m\in M\in {}_H^B{\mathcal{L}}_{\pi }$

$$\begin{array}{cc}\hfill & {(a\otimes x)}_{(1,\alpha )}{m}_{[-1,e]}{S}_{(H\otimes B)\alpha }\left({(a\otimes x)}_{(3,\alpha )}\right)\otimes {(a\otimes x)}_{(2,\alpha )}{▹}_{\alpha }{m}_{\left[0\right]}\hfill \\ \hfill =& a_{(1,2)}{R}^{\left(2\right)}{S}_H\left(a_{(1,3)}\right)\otimes {\sigma }_{\alpha ,e}(x_{(2,\alpha )},m_{(-1,e)(2,e)})\hfill \\ \hfill & x_{(1,\alpha )}{m}_{(-1,e)(1,e)}{S}_{B\alpha }\left(x_{(3,\alpha )}\right))\otimes a_{(1,2)}{R}^{\left(1\right)}·m_{(0,0)}\hfill \\ \hfill =& R^{\left(2\right)}\underline{a_{(1,2)}{S}_H\left(a_{(1,3)}\right)}\otimes {\sigma }_{\alpha ,e}(x_{(1,\alpha )},m_{(-1,e)(1,e)})m_{(-1,e)(2,e)}\hfill \\ \hfill \stackrel{\mathrm{by} \left(\mathrm{TQT3}\right) \mathrm{and} \left(\mathrm{TCT3}\right)}{=}& {\psi }_{e^{-1}}\left(x_{(2,\alpha )}{S}_{B\alpha }\left(x_{(3,\alpha )}\right)\right)\otimes R^{\left(1\right)}{a}_{(1,1)}·m_{(0,0)}\hfill \\ \hfill =& {\sigma }_{\alpha ,e}(x,m_{(-1,e)(1,e)})R^{\left(2\right)}\otimes m_{(-1,e)(2,e)}\otimes R^{\left(1\right)}a·m_{(0,0)}\hfill \\ \hfill \stackrel{\mathrm{(11)}}{=}& {\sigma }_{\alpha ,e}(x,m_{(-1,e)}){\rho }_e^M(a·m_{(0,0)})\hfill \\ \hfill =& {\rho }_e^M\left((a\otimes x){▹}_{\alpha }m\right).\hfill \end{array}$$

Then it follows from Theorem 2 for the braiding on ${}_{H\otimes B}^{H\otimes B_e}\mathcal{YD}$ that the braiding on ${}_H^B\mathcal{L}$ is the following:

$$\begin{array}{cc}\hfill & {\tau }_{M,N}(m\otimes n)\hfill \\ \hfill =& {}^{\alpha }\left(n_{[0,0]}\right)\otimes {\psi }_{\alpha }{S}_{\beta }^{-1}\left(n_{[-1,\beta ]}\right){▹}_{\alpha {\beta }^{-1}{\alpha }^{-1}}m\hfill \\ \hfill =& {\sigma }_{\alpha {\beta }^{-1}{\alpha }^{-1},\alpha }({\psi }_{\alpha }{S}_{\beta }^{-1}\left(n_{(-1,\beta )}\right),m_{(-1,\alpha )})R^{\left(1\right)}·n_{(0,0)}\otimes S_1^{-1}\left(R^{\left(2\right)}\right)·m_{(0,0)}.\hfill \end{array}$$

for all $M\in {}_H^B{\mathcal{L}}_{\alpha },N\in {}_H^B{\mathcal{L}}_{\beta }$, $m\in M$ and $n\in N$.

Furthermore, if $(H,R)$ is G-triangular and $(B,\sigma )$ is G-cotriangular, then we have:

$$\begin{array}{cc}\hfill & ({\tau }_{{}^{M}N,M}\circ {\tau }_{M,N})(m\otimes n)\hfill \\ \hfill =& {\sigma }_{\alpha {\beta }^{-1}{\alpha }^{-1},\alpha }({\psi }_{\alpha }{S}_{\beta }^{-1}\left(n_{(-1,\beta )}\right),m_{(-1,\alpha )}){\tau }_{{}^{M}N,M}(R^{\left(1\right)}·n_{(0,0)}\otimes S_1^{-1}\left(R^{\left(2\right)}\right)·m_{(0,0)})\hfill \\ \hfill =& {\sigma }_{\alpha {\beta }^{-1}{\alpha }^{-1},\alpha }({\psi }_{\alpha }{S}_{\beta }^{-1}\left(n_{(-1,\beta )}\right),m_{(-1,\alpha )})\hfill \\ \hfill & {\sigma }_{\left(\alpha \beta {\alpha }^{-1}\right){\alpha }^{-1}\left(\alpha {\beta }^{-1}{\alpha }^{-1}\right),\alpha \beta {\alpha }^{-1}}({\psi }_{\alpha \beta {\alpha }^{-1}}{S}_{\alpha }^{-1}\left(m_{(0,0)(-1,\alpha )}\right),{\psi }_{\alpha }\left(n_{(0,0)(-1,\beta )}\right)\hfill \\ \hfill & r^{\left(1\right)}{S}_1^{-1}\left(R^{\left(2\right)}\right)·m_{(0,0)(0,0)}\otimes S_1^{-1}\left(r^{\left(2\right)}\right)R^{\left(1\right)}·n_{(0,0)(0,0)}\hfill \\ \hfill =& {\sigma }_{\alpha {\beta }^{-1}{\alpha }^{-1},\alpha }({\psi }_{\alpha }{S}_{\beta }^{-1}\left(n_{(-1,\beta )(1,\beta )}\right),m_{(-1,\alpha )(1,\alpha )})\hfill \\ \hfill & {\sigma }_{\left(\alpha \beta {\alpha }^{-1}\right){\alpha }^{-1}\left(\alpha {\beta }^{-1}{\alpha }^{-1}\right),\alpha \beta {\alpha }^{-1}}({\psi }_{\alpha \beta {\alpha }^{-1}}{S}_{\alpha }^{-1}\left(m_{(-1,\alpha )(2,\alpha )}\right),{\psi }_{\alpha }\left(n_{(-1,\beta )(2,\beta )}\right)\hfill \\ \hfill & m_{(0,0)}\otimes n_{(0,0)}\hfill \\ \hfill \stackrel{\left(\mathrm{TCT4}\right)}{=}& {\sigma }_{\alpha {\beta }^{-1}{\alpha }^{-1},\alpha }({\psi }_{\alpha }{S}_{\beta }^{-1}\left(n_{(-1,\beta )(1,\beta )}\right),m_{(-1,\alpha )(1,\alpha )})\hfill \\ \hfill & {\sigma }_{{\alpha }^{-1},\alpha \beta {\alpha }^{-1}}(S_{\alpha }^{-1}\left(m_{(-1,\alpha )(2,\alpha )}\right),{\psi }_{\alpha {\beta }^{-1}}\left(n_{(-1,\beta )(2,\beta )}\right)m_{(0,0)}\otimes n_{(0,0)}\hfill \\ \hfill =& {\sigma }_{\alpha {\beta }^{-1}{\alpha }^{-1},\alpha }({\psi }_{\alpha }{S}_{\beta }^{-1}\left(n_{(-1,\beta )(1,\beta )}\right),m_{(-1,\alpha )(1,\alpha )})\hfill \\ \hfill & {\sigma }_{\alpha ,\alpha \beta {\alpha }^{-1}}^{-1}(m_{(-1,\alpha )(2,\alpha )},{\psi }_{\alpha {\beta }^{-1}}\left(n_{(-1,\beta )(2,\beta )}\right)m_{(0,0)}\otimes n_{(0,0)}\hfill \\ \hfill =& {\sigma }_{\alpha {\beta }^{-1}{\alpha }^{-1},\alpha }({\psi }_{\alpha }{S}_{\beta }^{-1}\left(n_{(-1,\beta )(1,\beta )}\right),m_{(-1,\alpha )(1,\alpha )})\hfill \\ \hfill & {\sigma }_{\alpha \beta {\alpha }^{-1},\alpha }({\psi }_{\alpha \beta {\alpha }^{-1}}{\psi }_{\alpha {\beta }^{-1}}(n_{(-1,\beta )(2,\beta )},m_{(-1,\alpha )(2,\alpha )})m_{(0,0)}\otimes n_{(0,0)}\hfill \\ \hfill \stackrel{\left(\mathrm{TCT1}\right)}{=}& {\sigma }_{1,\alpha }({\psi }_{\alpha }(n_{(-1,\beta )(2,\beta )}{S}_{\beta }^{-1}\left(n_{(-1,\beta )(1,\beta )}\right),m_{(-1,\alpha )})m_{(0,0)}\otimes n_{(0,0)}\hfill \\ \hfill =& m\otimes n.\hfill \end{array}$$

This proves that ${}_H^B\mathcal{L}$ is symmetric. □

Corollary 3.

Let $(B,\sigma )$ be a coquasitriangular π-graded Hopf algebra with crossing ψ. Then the category ${}^B{\mathbf{Mod}}_{\pi }={⨁}_{\alpha \in \pi }{}^B{\mathbf{Mod}}_{\alpha }$ is a braided monoidal subcategory of ${}_H^B\mathcal{L}$. The braiding on ${}^B{\mathbf{Mod}}_{\pi }$: ${\tau }_{M,N}:M\otimes N⟶N\otimes M$ is given by:

$${\tau }_{M,N}(m\otimes n)={\sigma }_{\alpha {\beta }^{-1}{\alpha }^{-1},\alpha }({\psi }_{\alpha }{S}_{\beta }^{-1}\left(n_{(-1,\beta )}\right),m_{(-1,\alpha )})n_{(0,0)}\otimes m_{(0,0)}$$

for all $M\in {}^B{\mathbf{Mod}}_{\alpha },N\in {}^B{\mathbf{Mod}}_{\beta }$, $m\in M$ and $n\in N$. Furthermore, if $(B,\sigma )$ is π-cotriangular, then ${}^B{\mathbf{Mod}}_{\pi }$ is symmetric.

6.2. Double By-Products

Let $(H,R)$ be a quasitriangular Hopf algebra and $(B,\sigma )$ coquasitriangular Hopf algebra. Then the category ${}_H^B\mathcal{L}$ is braided. Furthermore, if $(H,R)$ is triangular and $(B,\sigma )$ cotriangular, then ${}_H^B{\mathcal{L}}_{\pi }$ is symmetric.

Definition 11.

Let A be a braided Hopf algebra in ${}_H^B\mathcal{L}$. A double by-product $A★H★B$ is the set $A\otimes H\otimes B$ as a vector space, with multiplication and comultiplication respectively defined by

$$\begin{array}{ccc}& & (a★h★x)(b★l★y)=<x_1\mid b_{(-1)}>a(h_1·b_0)★h_{2}l★x_{2}y\hfill \\ & & \tilde{\Delta }(a★h★x)=a_1★R^{\left(2\right)}{h}_1★a_{2(-1)}{x}_1\otimes R^{\left(1\right)}·a_{20}★h_2★x_2\hfill \\ & & \widetilde{ϵ}(a\otimes h\otimes x)=ϵ\left(a\right)ϵ\left(h\right)ϵ\left(x\right),\hfill \end{array}$$

for any $a,b\in A;h,l\in H$ and $x,y\in B$.

Theorem 8.

Let A be a braided Hopf algebra in ${}_H^B\mathcal{L}$. Then the double by-product $A★H★B$ is a Hopf algebra. Its antipode is given via

$$\tilde{S}(a★h★x)=(1_A★1_H★S_B\left(a_{(-1)}x\right))(S_H\left(R^{\left(2\right)}{}_2{h}_2\right)·S_A(R^{\left(1\right)}·a_0)★S_H\left(R^{\left(2\right)}{}_1{h}_1\right)★1_B),$$

for any $a\in A;h\in H$ and $x\in B$.

Proof. 

Let A be a braided Hopf algebra in ${}_H^B\mathcal{L}$. Since A is an algebra in ${}_H\mathbf{Mod}$, by Radford’s by-product theorem in [10], we have a Hopf algebra $A★H$ where it has a multiplication and comultiplication given by

$$\begin{array}{ccc}& & (a★h)(b★l)=a(h_1·b)★h_{2}l\hfill \\ & & \tilde{\Delta }(a★h)=a_1★R^{\left(2\right)}{h}_1\otimes R^{\left(1\right)}·a_{20}★h_2\hfill \\ & & \widetilde{ϵ}(a\otimes h)=ϵ\left(a\right)ϵ\left(h\right)\hfill \end{array}$$

for any $a,b\in A$ and $h,l\in H$.

Define a left coaction of B on $A★H$ and a left action of B on $A★H$ respectively as follows:

$${\rho }_{A★H}(a★h)=a_{(-1)}\otimes a_0★h$$

and

$$x·(a★h)=<x\mid a_{(-1)}>a_0★h,$$

for any $a\in A;h\in H$ and $x\in B$.

It is easy to see that $(A★H,{\rho }_{A★H})$ is a left B-comodule. Moreover, $A★H$ is a left B-comodule algebra. In fact, we have

$$\begin{array}{cc}\hfill & {\rho }_{A★H}\left((a★h)(b★l)\right)={\left(a(h_1·b)\right)}_{(-1)}\otimes {\left(a(h_1·b)\right)}_0★h_{2}l\hfill \\ \hfill =& a_{(-1)}{(h_1·b)}_{(-1)}\otimes a_0{(h_1·b)}_0★h_{2}l=a_{(-1)}{b}_{(-1)}\otimes a_0(h_1·b_0)★h_{2}l\hfill \\ \hfill =& a_{(-1)}{b}_{(-1)}\otimes (a_0★h)(b_0★l)={\rho }_{A★H}(a★h){\rho }_{A★H}(b★l)\hfill \end{array}$$

for any $a\in A;h\in H$ and $x\in B$.

Similarly, we can check that $A★H$ is also a left B-comodule co-algebra. Then this shows that $A★H$ is a left B-comodule bi-algebra in ${}_H^B\mathcal{L}$. Thus, again by Radford’ by-product theorem, we have a Hopf algebra $A★H★B$.

Finally, we have

$$\begin{array}{cc}\hfill & \tilde{S}(a★h★x)=(1_A★1_H★S_B\left({(a★h)}_{(-1)}x\right))(S_{A★H}{(a★h)}_0★1_B)\hfill \\ \hfill =& (1_A★1_H★S_B\left(\left(a_{(-1)}x\right)\right)(S_{A★H}(a_0★h)★1_B)\hfill \\ \hfill =& (1_A★1_H★S_B\left(\left(a_{(-1)}x\right)\right)((1★S_H\left(R^{\left(2\right)}h\right)(S_A(R^{\left(1\right)}·a_0)★1_H)★1_B)\hfill \\ \hfill =& (1_A★1_H★S_B\left(\left(a_{(-1)}x\right)\right)(S_H\left(R^{\left(2\right)}{}_2{h}_2\right)·S_A(R^{\left(1\right)}·a_0)★S_H\left(R^{\left(2\right)}{}_1\right)★1_B)\hfill \\ \hfill =& (1_A★1_H★S_B\left(a_{(-1)}x\right))(S_H\left(R^{\left(2\right)}{}_2{h}_2\right)·S_A(R^{\left(1\right)}·a_0)★S_H\left(R^{\left(2\right)}{}_1{h}_1\right)★1_B),\hfill \end{array}$$

as required. □

Let $V\in {}_H^B\mathcal{L}$ be a finite-dimensional vector space over $\mathbb{k}$. Let $\Phi$: $H⟶EndV$ denote the corresponding representation. H then acts diagonally on $V^{\otimes n}$, and we have the resulting representation ${\Phi }^n={\Phi }^{\otimes n}\circ {\Delta }_{n-1}:H⟶End{V}^{\otimes n}$, such that for all $v_1,\cdots ,v_n\in V$.

$${\Phi }^n\left(h\right)(v_1\otimes \cdots \otimes v_n)=\sum h_1·v_1\otimes \cdots \otimes h_n·v_n.$$

For the Hopf algebra $A★H★B$, one lets $\Phi (a★1★1)\left(v\right)=a·v$, $\Phi (1★h★1)\left(v\right)=h·v$, $\Phi (1★1★x)\left(v\right)=<x\mid v_{(-1)}>v_0$, and $\Phi (a★h★x)=\Phi (a★1★1)\Phi (1★h★1)\Phi (1★1★x)$, for any $a\in A$, $h\in H$, $x\in B$ and $v\in V$. It is easy to check that $\Phi$ is a representation of Hopf algebras $A★H★B$.

For $u\in A★H★B$, we denote $\Phi \left(u\right)$ by $\underline{u}$, in particular, if $a\in A$ then one denotes it by $\underline{a}$. Hence, we have a representation via:

$${\Phi }^n\left(u\right)(v_1\otimes \cdots \otimes v_n)={\underline{u}}_1·\left(v_1\right)\otimes \cdots \otimes {\underline{u}}_n·\left(v_n\right),$$

where ${\Delta }^{m-1}\left(u\right)=u_1\otimes \cdots \otimes u_m$.

Proposition 8.

Let A be a bi-algebra in ${}_H^B\mathcal{L}$. If the elements of A are $(1,1)$-primitive, i.e., $\Delta \left(a\right)=a\otimes 1+1\otimes a$ for all $a\in A$, then

(i) the following hold: $\forall a\in A$

$$\begin{array}{ccc}& & {\Delta }_m(a★1★1)=(a★1★1)\otimes {(1★1★1)}^m\hfill \\ & & +\sum _{j=1}^m{\Delta }_{j-1}(1★R^{\left(2\right)}★a_{(-1)})\otimes (R^{\left(1\right)}·a_0★1★1)\otimes {(1★1★1)}^{m-j};\hfill \end{array}$$

(ii) ${\sigma }_1·(a\otimes 1)={\Phi }^2\left[(1★R^{\left(2\right)}★a_{(-1)})(R^{\left(1\right)}·a_0★1★1)\right]$;

(iii) ${\sigma }_m{\sigma }_{m-1}\cdots {\sigma }_1·(\underline{a}\otimes {\underline{1}}^{\otimes m})={\Delta }_{m-1}\underline{(R^{\left(2\right)}★a_{(-1)})}\otimes R^{\left(1\right)}·\underline{a_0}$;

(iv) ${\Phi }^{m+1}(a★1★1)=({\sum }_{i=0}^m{\sigma }_i{\sigma }_{i-1}\cdots {\sigma }_1{\sigma }_0)(\underline{a}\otimes 1^{\otimes m})$.

Proof. 

Straightforward. □

By the previous results, we have the following results:

Proposition 9.

Let A be a braided-co-commutative Hopf algebra in ${}_H^B\mathcal{L}$, and V be an A-module in ${}_H{\mathcal{L}}^B$. Then the actions of $A★H★B$ and $\mathbb{k}{S}_n$ on $V^{\otimes n}$ commute. In particular, $K=U\left(End{\left(V\right)}^{-}\right)★H★B$ and $\mathbb{k}{S}_n$ centralize each other in $End{V}^{\otimes n}$.

Theorem 9.

Let $(H,R)$ be a triangular Hopf algebra and $(B,\sigma )$ cotriangular π-graded Hopf algebra with crossing ψ. Let V be an object of ${}_H^B\mathcal{L}$ with finite dimension. Construct the Hopf algebra $K=U\left({\left(En{d}_{\mathbb{k}}V\right)}^{-}\right)★H★B$. If $n!\in {\mathbb{k}}^{\times }$, then

(i)

$En{d}_{\Psi \left(\mathbb{k}{S}_n\right)}{V}^{\otimes n}={\Phi }^n\left(K\right)$,

(ii)

$En{d}_{{\Phi }^n\left(K\right)}{V}^{\otimes n}=\Psi \left(\mathbb{k}{S}_n\right)$.

6.3. An Example of Double By-Product

An example of a double by-product is studied in this subsection.

Definition 12.

An algebra $M_q\left(2\right)$ is generated by four elements $a,b,c,d$ satisfying the following six relations:

$$\begin{array}{ccc}& & ba=ab,bc=cb,bd=qdb,\hfill \\ & & ca=ac,da=ad,cd=qdc.\hfill \end{array}$$

Lemma 3.

Given an algebra J. Let $A,B,C$ and D be four elements of J. If the quadruple $(A,B,C,D)$ satisfy the relations:

$$\begin{array}{ccc}& & BA=AB,BC=CB,BD=qDB,\hfill \\ & & CA=AC,DA=AD,CD=qDC,\hfill \end{array}$$

then there exists an algebra homomorphism ϕ: $M_q\left(2\right)⟶J$ such that $\varphi \left(a\right)=A,\varphi \left(b\right)=B$, $\varphi \left(c\right)=c$ and $\varphi \left(d\right)=D$.

Proposition 10.

The algebra $M_q\left(2\right)$ is Noetherian and has no zero divisors. A basis for the underlying vector space is given by the set of monomials $\{a^k{b}^l{c}^m{d}^n\mid k,l,m,n\in N\}$.

Proof. 

We will prove this proposition by building a tower of algebras

$$A_0=\mathbb{k}\subset A_1\subset A_2\subset A_3\subset A_4=M_q\left(2\right).$$

Define the algebras $A_1=\mathbb{k}\left[a\right]$, $A_2=\mathbb{k}\{A,b\}/(ba-ab)$, $A_3=\mathbb{k}\{a,b,c\}/(ba-ab,ca-ac,cb-bc)$. It is obvious that $A_0=\mathbb{k}\subset A_1\subset A_2\subset A_3$ are all trivially Ore extensions.

Define $\alpha$: $A_3⟶A_3$ via $a\mapsto a$, $b\mapsto q^{-1}b$, $c\mapsto q^{-1}c$, it is easy to see that $\alpha$ is an automorphism of algebra $A_3$.

Let $\delta =0:A_3⟶A_3$ be a trivial map. Then it is not hard to show that an Ore extension $A_3[d,\alpha ,0]$ of the algebra $A_3$ is isomorphic to $M_q\left(2\right)$, i.e., $A_3[d,\alpha ,0]\cong M_q\left(2\right)$.

It is also obvious that $\{a^k{b}^l{c}^m{d}^n\mid k,l,m,n\in N\}$ is a basis for the underlying vector space of $M_q\left(2\right)$.

This finishes the proof. □

Theorem 10.

The relations:

$$\begin{array}{ccc}& & \Delta \left(a\right)=a\otimes a,\Delta \left(b\right)=b\otimes b,\hfill \\ & & \Delta \left(c\right)=c\otimes c,\Delta \left(d\right)=b\otimes d+d\otimes a,\hfill \\ & & ϵ\left(a\right)=ϵ\left(b\right)=ϵ\left(c\right)=1,ϵ\left(d\right)=0\hfill \end{array}$$

define two algebra homomorphism Δ: $M_q\left(2\right)⟶M_q\left(2\right)\otimes M_q\left(2\right)$ and ϵ: $M_q\left(2\right)⟶\mathbb{k}$. Then $M_q\left(2\right)$ is a non-commutative and non-co-commutative bi-algebra.

Moreover,

$$\Delta \left(a^k{b}^l{c}^m{d}^n\right)=\sum _{-1<s<n+1}{\left(\genfrac{}{}{0pt}{}{n}{s}\right)}_q{a}^k{b}^{l+s}{c}^m{d}^{n-s}\otimes a^{n+k-s}{b}^l{c}^m{d}^s.$$

Proof. 

In order to show that $(\Delta \otimes id)\Delta =(id\otimes \Delta )\Delta$, it is sufficient to check it for the generators $a,b,c,d$. However, a routine verification can finish the proof.

By a computation, one has

$$\Delta \left(a^k{b}^l{c}^m{d}^n\right)=\sum _{-1<s<n+1}{\left(\genfrac{}{}{0pt}{}{n}{s}\right)}_q{a}^k{b}^{l+s}{c}^m{d}^{n-s}\otimes a^{n+k-s}{b}^l{c}^m{d}^s.$$

This completes the proof. □

We have that $(\mathbb{k}{Z}_2,R)$ is a triangular Hopf algebra with $R=1/2(1\otimes 1+1\otimes g+g\otimes 1+g\otimes g)$, and $(B,<\mid >)$ is a cotriangular Hopf algebra with $<1\mid g>=1=<1\mid 1>=<g\mid 1>;<g\mid g>=-1$.

Let $A=\mathbb{k}\left[x\right]$. We consider A in ${}_H\mathbf{Mod}$ by $1·x^j=x^j,g·x^j={(-1)}^j{x}^j$; one considers A in ${}^B\mathbf{Mod}$ by $\rho \left(x^j\right)=g^{\left[j\right]}\otimes x^j$, where $\left[j\right]\equiv j\left(2\right)$. It is easy to see that A is an algebra in ${}_H^B\mathcal{L}$, and ${\tau }_{A,A}(x^l\otimes x^m)=x^m\otimes x^l$. So we have

Proposition 11.

With notation above, we have that $A\#H\#B\cong M_{-1}\left(2\right)$ as Hopf algebra.

Proof. 

It is easy to see that the set $\{1\#1\#1,1\#g\#1,x\#1\#1,1\#1\#G\}$ forms a basis of the double by-product $A\#H\#B$.

Define $\Phi$: $A\#H\#B⟶M_{-1}\left(2\right)$ by

$$1\#1\#1\mapsto a;1\#1\#G\mapsto b;1\#g\#1\mapsto c;x\#1\#1\mapsto d.$$

Straightforward. This finishes the proof. □

6.4. Further Research

In this paper, we have studied $\pi$-graded Hopf algebras, as well as new non-trivial examples of symmetric braided monoidal category. are constructed. We have established a $\pi$-double centralizer property in the $\pi$-graded Hopf algebra framework (see [15,16]). It is still not clear if the nicer results, obtained in the $\pi$-graded case, can be pushed forward to the multiplier Hopf algebra case so that better results can also be shown.