A Duality Theorem for Hopf Quasimodule Algebras

Abstract

In this paper, we introduce and study two smash products $A★H$ for a left H-quasimodule algebra A over a Hopf quasigroup H over a field $\mathbb{K}$ and $B\#U$ for a coquasi U-module algebra B over a Hopf coquasigroup U, respectively. Then, we prove our duality theorem $(A★H)\#H^{*}\cong A\otimes (H\#H^{*})\cong A\otimes M_n\left(\mathbb{K}\right)\cong M_n\left(A\right)$ in the setting of a Hopf quasigroup H of dimension n. As an application of our result, we consider a special case of a finite quasigroup.

1. Introduction

The notion of Hopf quasigroup was introduced by Klim and Majid in [1], which is a particular case of the notion of unital counital coassociative bialgebra introduced in [2]. Dually, the vector space of linear functionals on a finite quasigroup carries the structure of a Hopf coquasigroup (cf. [1]), which is a counital unital associative bialgebra. These Hopf quasigroups and Hopf coquasigroups are generalizations of Hopf algebras (see [3]). These notions are related to cohomology modules [4], Yetter–Drinfeld Modules [5,6,7,8], and coalgebras [9] based on digital images.

Given a locally compact abelian group G and a von Neumann algebra N, let G act on N via a homomorphism $\alpha$ of G into $Aut\left(N\right)$. Then, we have the smash product algebra $N{\times }_{\alpha }G$. Takesaki in 1973 introduced an action $\widehat{\alpha }$ of the dual group $\widehat{G}$ and proved the duality theorem: $\left(N{\times }_{\alpha }G\right){\times }_{\widehat{\alpha }}\widehat{G}\cong N\overline{\otimes }M$ (as algebras), where M is the algebra of all bounded operators on $L_2\left(G\right)$ (see [10]). This result was extended to arbitrary locally compact groups G independently by Landstad and Nakagami in the mid 1970s (see [11,12]).

In the above duality theorem, if we replace G by a Hopf algebra and N by an algebra, then the authors in [13] constructed a duality theorem for any Hopf algebra H-module algebra A over a field $\mathbb{K}$ under some condition “locally finite”. In particular, if H is finite-dimensional, then the duality theorem is

$$(A\#H)\#H^{*}\cong A\otimes (H\#H^{*})\cong A\otimes M_n\left(\mathbb{K}\right)\cong M_n\left(A\right).$$

It is now natural to ask whether the duality theorem above in [13] holds in the framework of Hopf quasigroups. This becomes our motivation of writing this paper. We will overcome non-associativity in Hopf quasigroups and non-coassociativity in Hopf coquasigroups by introducing some new notions and developing new ways.

This article is organized as follows: In Section 2, we recall and investigate some basic definitions and properties related to Hopf (co)quasigroups.

In Section 3, we introduce and study two smash products $A★H$ for a left H-quasimodule algebra A over a Hopf quasigroup H and $B\#U$ for a coquasi U-module algebra B over a Hopf coquasigroup U, respectively. In Section 4, we prove our duality theorem in the setting of finite-dimensional Hopf quasigroups (see Theorem 1). As an application of our result, we consider a special case of a finite quasigroup.

Throughout this paper, $\mathbb{K}$ is a fixed field, and all vector spaces are over $\mathbb{K}$. By linear maps, we mean $\mathbb{K}$-linear maps. Unadorned ⊗ means ${\otimes }_{\mathbb{K}}$. Let C be a coalgebra with a coproduct $\mathrm{\Delta }$. We will use the Heyneman–Sweedler’s notation (see [3]), $\mathrm{\Delta }\left(c\right)=\sum c_{\left(1\right)}\otimes c_{\left(2\right)}$ for all $c\in C$, for coproduct.

2. Preliminaries

In this section, some basic definitions and properties of Hopf (co)quasigroups and smash products are recalled and investigated.

2.1. Algebras and Coalgebras

The following notions can be found in [2]. An algebra $(A,\nabla )$ is a vector space A equipped with a linear map $\nabla :A\otimes A⟶A$. The algebra $(A,\nabla )$ is called associative if $\nabla (id\otimes \nabla )=\nabla (\nabla \otimes id)$. It is customary to write $\nabla (x\otimes y)=xy,\forall x,y\in A$. A unital algebra$(A,\nabla ,\mu )$ is a vector space A equipped with two linear maps $\nabla :A\otimes A⟶A$ and $\mu :\mathbb{K}⟶A$ such that $\nabla (id\otimes \mu )=id=\nabla (\mu \otimes id)$. Generally, we write $1\in A$ for $\mu \left(1_{\mathbb{K}}\right)$.

Dually, a coalgebra $(C,\mathrm{\Delta })$ is a vector space C equipped with a linear map $\mathrm{\Delta }:C⟶C\otimes C$. The coalgebra $(C,\mathrm{\Delta })$ is called coassociative if $(id\otimes \mathrm{\Delta })\mathrm{\Delta }=(\mathrm{\Delta }\otimes id)\mathrm{\Delta }$. A counital coalgebra $(C,\mathrm{\Delta },\epsilon )$ is a vector space C equipped with two linear maps $\mathrm{\Delta }:C⟶C\otimes C$ and $\epsilon :C⟶\mathbb{K}$ such that $(id\otimes \epsilon )\mathrm{\Delta }=id=(\epsilon \otimes id)\mathrm{\Delta }$.

A bialgebra $(A,\nabla ,\mathrm{\Delta })$ is an algebra $(A,\nabla )$ and a coalgebra $(A,\mathrm{\Delta })$ such that $\mathrm{\Delta }\left(xy\right)=\mathrm{\Delta }\left(x\right)\mathrm{\Delta }\left(y\right)$ for all $x,y\in A$. A unital bialgebra$(A,\nabla ,\mu ,\mathrm{\Delta })$ is a coalgebra $(A,\mathrm{\Delta })$ and a unital $(A,\nabla ,\mu )$ such that $\mathrm{\Delta }\left(xy\right)=\mathrm{\Delta }\left(x\right)\mathrm{\Delta }\left(y\right)$ and $\mathrm{\Delta }\left(1\right)=1$ for all $x,y\in A$. A counital bialgebra $(A,\nabla ,\mathrm{\Delta },\epsilon )$ is a counital coalgebra $(A,\mathrm{\Delta },\epsilon )$ and an algebra $(A,\nabla )$ such that $\mathrm{\Delta }\left(xy\right)=\mathrm{\Delta }\left(x\right)\mathrm{\Delta }\left(y\right)$ and $\epsilon \left(xy\right)=\epsilon \left(x\right)\epsilon \left(y\right)$ for all $x,y\in A$. A unital counital bialgebra $(A,\mathrm{\Delta },\epsilon ,\nabla ,\mu )$ is both a unital bialgebra $(A,\mathrm{\Delta },\nabla ,\mu )$ and a counital bialgebra $(A,\mathrm{\Delta },\epsilon ,\nabla )$ such that $\epsilon \left(1\right)=1$. A Hopf algebra always means a unital counital associative coassociative bialgebra with an antipode (cf. [3]).

2.2. Hopf (Co)quasigroups

Recall from [1] that a Hopf quasigroup is a unital counital coassociative bialgebra $(H,\nabla ,\mu ,\mathrm{\Delta },\epsilon )$ armed with a linear map $S:H⟶H$ (called antipode) such that

$$\begin{array}{ccc}& & \sum S\left(h_{\left(1\right)}\right)\left(h_{\left(2\right)}g\right)=\epsilon \left(h\right)g=\sum h_{\left(1\right)}\left(S\left(h_{\left(2\right)}\right)g\right),\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & \sum \left(h{g}_{\left(1\right)}\right)S\left(g_{\left(2\right)}\right)=h\epsilon \left(g\right)=\sum \left(hS\left(g_{\left(1\right)}\right)\right)g_{\left(2\right)}\hfill \end{array}$$

(2)

for any $h,g\in H$.

Dually, a Hopf coquasigroup is a counital unital associative bialgebra $(H,\nabla ,\mu ,\mathrm{\Delta },\epsilon )$ equipped with a linear map $S:H⟶H$ (called antipode) such that

$$\begin{array}{ccc}& & \sum S\left(h_{\left(1\right)}\right)h_{\left(2\right)\left(1\right)}\otimes h_{\left(2\right)\left(2\right)}=1\otimes h=\sum h_{\left(1\right)}S\left(h_{\left(2\right)\left(1\right)}\right)\otimes h_{\left(2\right)\left(2\right)},\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & \sum h_{\left(1\right)\left(1\right)}\otimes S\left(h_{\left(1\right)\left(2\right)}\right)h_{\left(2\right)}=h\otimes 1=\sum h_{\left(1\right)\left(1\right)}\otimes h_{\left(1\right)\left(2\right)}S\left(h_{\left(2\right)}\right)\hfill \end{array}$$

(4)

for all $h\in H$.

The following remark is helpful to compute something later.

Remark 1.

Let H be a Hopf quasigroup or a coquasigroup with antipode S.

(1) 

A Hopf (co)quasigroup is a Hopf algebra if and only its (co)product is (co)associative;

(2) 

About S, we have

$$\begin{array}{ccc}& & \sum S\left(h_{\left(1\right)}\right)h_{\left(2\right)}=\sum h_{\left(1\right)}S\left(h_{\left(2\right)}\right)=\epsilon \left(h\right)1,\hfill \\ & & S\left(hg\right)=S\left(g\right)S\left(h\right),\mathrm{\Delta }\left(S\left(h\right)\right)=\sum S\left(h_{\left(2\right)}\right)\otimes S\left(h_{\left(1\right)}\right)\hfill \end{array}$$

for all $h,g\in H$;

(3) 

If S is a bijective antipode S with an inverse $S^{-1}$, then

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

for all $h,g\in H$.

If H is a finite dimensional Hopf quasigroup with antipode S, then its linear dual $H^{*}$ is not Hopf quasigroup but a Hopf coquasigroup with antipode $S^{*}$, and one has the non-degenerate bilinear form

$$⟨,⟩:H^{*}\times H⟶\mathbb{K}$$

given by $⟨h^{*},h⟩=h^{*}\left(h\right)$ for all $h^{*}\in H^{*}$ and $h\in H$. Let $h^{*}\in H^{*}$ and $h\in H$. Then, we have $⟨S^{*}\left(h^{*}\right),h⟩=⟨h^{*},S\left(h\right)⟩$, and the left action of $h^{*}$ on h (denoted by $h^{*}\rightharpoonup h$) is given by

$$h^{*}\rightharpoonup h=\sum ⟨h^{*},h_{\left(2\right)}⟩h_{\left(1\right)}.$$

(5)

Similarly the right action of $h^{*}$ on h is denoted by $h\leftharpoonup h^{*}$ and is given by

$$h\leftharpoonup h^{*}=\sum ⟨h^{*},h_{\left(1\right)}⟩h_{\left(2\right)}.$$

(6)

Proposition 1.

Let H be a finite dimensional Hopf quasigroup. Let $h^{*},l^{*}\in H$ and $h\in H$. Then

(a) 

$h^{*}\rightharpoonup (l^{*}\rightharpoonup h)=\left(h^{*}{l}^{*}\right)\rightharpoonup h$;

(b) 

$(h\leftharpoonup h^{*})\leftharpoonup l^{*}=h\leftharpoonup \left(h^{*}{l}^{*}\right)$.

Proof .

(a) We compute:

$$\begin{array}{ccc}\hfill h^{*}& \rightharpoonup & (l^{*}\rightharpoonup h)=\sum ⟨l^{*},h_{\left(2\right)}⟩⟨h^{*},h_{\left(1\right)\left(2\right)}⟩h_{\left(1\right)\left(1\right)}\hfill \\ & =& \sum ⟨l^{*},h_{\left(3\right)}⟩⟨h^{*},h_{\left(2\right)}⟩h_{\left(1\right)}\hfill \\ & =& \sum ⟨h^{*}{l}^{*},h_{\left(2\right)}⟩h_{\left(1\right)}=\left(h^{*}{l}^{*}\right)\rightharpoonup h;\hfill \end{array}$$

(b) follows similarly. □

Proposition 2.

Let H be a finite dimensional Hopf quasigroup. Let $h^{*}\in H$ and $h,l\in H$. Then

(a) 

$\mathrm{\Delta }(h^{*}\rightharpoonup h)=\sum h_{\left(1\right)}\otimes (h^{*}\rightharpoonup h_{\left(2\right)})$;

(b) 

$\mathrm{\Delta }(h\leftharpoonup h^{*})=\sum (h_{\left(1\right)}\leftharpoonup h^{*})\otimes h_{\left(2\right)}$;

(c) 

$h^{*}\rightharpoonup \left(hl\right)=\sum (h_{\left(1\right)}^{*}\rightharpoonup h)(h_{\left(2\right)}^{*}\rightharpoonup l)$;

(d) 

$\left(hl\right)\leftharpoonup h^{*}=\sum (h\leftharpoonup h_{\left(1\right)}^{*})(l\leftharpoonup h_{\left(2\right)}^{*})$.

Proof .

(a) We compute:

$$\begin{array}{ccc}\hfill \mathrm{\Delta }(h^{*}\rightharpoonup h)& =& \sum ⟨h^{*},h_{\left(2\right)}⟩h_{\left(1\right)\left(1\right)}\otimes h_{\left(1\right)\left(2\right)}\hfill \\ & =& \sum ⟨h^{*},h_{\left(2\right)\left(2\right)}⟩h_{\left(1\right)}\otimes h_{\left(2\right)\left(1\right)}=\sum h_{\left(1\right)}\otimes (h^{*}\rightharpoonup h_{\left(2\right)});\hfill \end{array}$$

(b)–(d) follow similarly. □

2.3. Quasimodules

Let H be a Hopf quasigroup. The following notion is given in [5].

We say that $(M,▹)$ is a left H-quasimodule if M is a vector space and $▹:H\otimes M⟶M$ is a linear map (called the left quasiaction) satisfying $1▹m=m$ and

$$\sum h_{\left(1\right)}▹(S\left(h_{\left(2\right)}\right)▹m)=\sum S\left(h_{\left(1\right)}\right)▹(h_{\left(2\right)}▹m)=\epsilon \left(h\right)(l▹m)$$

(7)

for all $h\in H$ and $m\in M$.

Remark 2.

(1) If H has an invertible antipode S with an inverse $S^{-1}$, then

$$\sum S^{-1}\left(h_{\left(2\right)}\right)▹(h_{\left(1\right)}▹m)=\sum (h_{\left(2\right)})▹(S^{-1}\left(h_{\left(1\right)}\right)▹m)=\epsilon \left(h\right)(l▹m)$$

(8)

for all $h\in H$ and $m\in M$;

(2) Similarly, we can define a right H-quasimodule. We say that $(M,◃)$ is a right H-quasimoduleif M is a vector space, and $◃:M\otimes H⟶M$ is a linear map (called the rightquasiaction) satisfying $1◃m=m$ and

$$\sum (m◃h_{\left(1\right)})◃S\left(h_{\left(2\right)}\right)=\sum (m◃S\left(h_{\left(1\right)}\right))◃h_{\left(2\right)}=m\epsilon \left(h\right)$$

for all $h\in H,m\in M$.

Example 1.

Let H be a finite dimensional Hopf quasigroup.

(1) 

Then, $(H^{*},\rightharpoonup )$ is a left H-quasimodule. In fact, e.g., we have

$$\begin{array}{ccc}\hfill \sum h_{\left(1\right)}& \rightharpoonup & (S\left(h_{\left(2\right)}\right)\rightharpoonup h^{*})=\sum ⟨S\left(h_{\left(2\right)}\right),h_{\left(2\right)}^{*}⟩⟨h_{\left(1\right)},h_{\left(1\right)\left(2\right)}^{*}⟩h_{\left(1\right)\left(1\right)}^{*}\hfill \\ & =& \sum ⟨h_{\left(2\right)},S^{*}(h_{\left(2\right)}^{*})⟩⟨h_{\left(1\right)},h_{\left(1\right)\left(2\right)}^{*}⟩h_{\left(1\right)\left(1\right)}^{*}\hfill \\ & =& \sum ⟨h,h_{\left(1\right)\left(2\right)}^{*}{S}^{*}(h_{\left(2\right)}^{*})⟩h_{\left(1\right)\left(1\right)}^{*}\hfill \\ & \stackrel{\left(4\right)}{=}& \sum \epsilon \left(h\right)h^{*}.\hfill \end{array}$$

for any $h,l\in H$ and $h^{*}\in H^{*}$;

(2) 

Similarly, $(H^{*},\leftharpoonup )$ is a right H-quasimodule.

3. Two Smash Products

In this section, we will consider two smash products for Hopf quasigroups and Hopf coquasigroup in order to obtain our duality theorem.

3.1. Quasimodule Algebra over Hopf Quasigroup

Definition 1.

Let H be a Hopf quasigroup. Then,

(1) 

A unital algebra A is said to be a left H-quasimodule algebraif A is a left H-quasimodule such that, for all $a,b\in A$,

$$\begin{array}{ccc}& & h▹1_A=\epsilon \left(h\right)1_A;\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& & h▹\left(ab\right)=\sum (h_{\left(1\right)}▹a)(h_{\left(2\right)}▹b);\hfill \end{array}$$

(10)

(2) 

If A is a left H-quasimodule algebra, one can define a smash product $A★H=A\otimes H$ with a product given by

$$\begin{array}{c}\hfill (a★x)(b★y)=\sum a(x_{\left(1\right)}▹b)★x_{\left(2\right)}y\end{array}$$

(11)

for any $a,b\in A$ and $x,y\in H$.

Remark 3.

(1) It follows immediately from Equation (3) above that $(a★x)(1★y)=\sum a★xy$ and $(a★1)(b★y)=\sum ab★y$, for any $a,b\in A$ and $x,y\in H$;

(2) The study of smash product is also referred to the papers [14,15].

Example 2.

(1) Let H be a Hopf quasigroup. Then $\mathbb{K}$ is a left H-quasimodule algebra with the trivial action given by $h▹a=\epsilon \left(h\right)a$, for $h\in H,a\in \mathbb{K}$. Thus, we have $\mathbb{K}★H\cong H$;

(2) Let H be a finite-dimensional Hopf coquasigroup. Then, $H^{*}$ is a finite-dimensional Hopf quasigroup. A unital associative algebra H can be regarded as a left $H^{*}$-quasimodule algebra with ⇀. Hence, we can form smash product $H★H^{*}$ with the following product:

$$(h★h^{*})(l★l^{*})=\sum ⟨h_{\left(1\right)}^{*},l_{\left(2\right)}⟩h{l}_{\left(1\right)}★h_{\left(2\right)}^{*}{l}^{*}$$

(12)

for any $h,l\in A$ and $h^{*},l^{*}\in H^{*}$.

In fact, by Example 1, we just check (1) and (2). In fact, for any $h,l\in H$ and $h^{*}\in H^{*}$, it is obvious that $h^{*}\rightharpoonup 1_H=⟨h^{*},1_H⟩1_H$. For (2), we have

$$\begin{array}{ccc}\hfill h^{*}\rightharpoonup \left(hl\right)& =& \sum ⟨h^{*},h_{\left(2\right)}{l}_{\left(2\right)}⟩h_{\left(1\right)}{l}_{\left(1\right)}\hfill \\ & =& \sum ⟨h_{\left(1\right)}^{*},h_{\left(2\right)}⟩⟨h_{\left(2\right)}^{*},l_{\left(2\right)}⟩h_{\left(1\right)}{l}_{\left(1\right)}=(h_{\left(1\right)}^{*}\rightharpoonup h)(h_{\left(2\right)}^{*}\rightharpoonup l).\hfill \end{array}$$

Furthermore, it is easy to check that H is not a left $H^{*}$-module.

Proposition 3.

With notations as above, then, $A★H$ is a unital algebra with unit $1_A★1_H$. Furthermore, $A★H$ is an associative algebra if and only if H is a Hopf algebra, and A is the usual left H-module algebra.

Proof. 

Obviously, $A★H$ is a unital algebra with unit $1_A★1_H$. Furthermore, for any $a,b,\in A$ and $x,y,z\in H$, we have

$$\begin{array}{ccc}\hfill \left[\right(a★x\left)\right(b★y\left)\right](c★z)& =& \sum [a(x_{\left(1\right)}▹b)★x_{\left(2\right)}y](c★z)\hfill \\ & =& \sum \left[a(x_{\left(1\right)}▹b)\right][{\left(x_{\left(2\right)}y\right)}_{\left(1\right)}▹c]★{\left(x_{\left(2\right)}y\right)}_{\left(2\right)}z\hfill \\ & =& \sum \left[a(x_{\left(1\right)}▹b)\right][x_{\left(2\right)\left(1\right)}{y}_{\left(1\right)}▹c]★\left[x_{\left(2\right)\left(2\right)}{y}_{\left(2\right)}\right]z\hfill \\ & =& \sum \left[a(x_{\left(1\right)}▹b)\right][x_{\left(2\right)}{y}_{\left(1\right)}▹c]★\left[x_{\left(3\right)}{y}_{\left(2\right)}\right]z\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill (a★x)\left[\right(b★y\left)\right(c★z\left)\right]& =& \sum (a★x)[b(y_{\left(1\right)}▹c)★y_{\left(2\right)}z]\hfill \\ & =& \sum a[x_{\left(1\right)}▹\left(b(y_{\left(1\right)}▹c)\right)]★x_{\left(2\right)}\left[y_{\left(2\right)}z\right]\hfill \\ & =& \sum a\left[(x_{\left(1\right)\left(1\right)}▹b)(x_{\left(1\right)\left(2\right)}▹(y_{\left(1\right)}▹c))\right]★x_{\left(2\right)}\left[y_{\left(2\right)}z\right]\hfill \\ & =& \sum a\left[(x_{\left(1\right)}▹b)(x_{\left(2\right)}▹(y_{\left(1\right)}▹c))\right]★x_{\left(3\right)}\left[y_{\left(2\right)}z\right].\hfill \end{array}$$

If $A★H$ is associative, then we have

$$\left(x_{\left(1\right)}{y}_{\left(1\right)}\right)▹c\otimes \left[x_{\left(2\right)}{y}_{\left(2\right)}\right]z=(x_{\left(1\right)}▹(y_{\left(1\right)}▹c))\otimes x_{\left(2\right)}\left[y_{\left(2\right)}z\right]$$

by taking $a=b=1$, and

$$\left(ab\right)c=a\left(bc\right)$$

by taking $x=y=z=1$. It is easy to obtain that H is a Hopf algebra, and A is the usual left H-module algebra.

Conversely, it is obvious. □

3.2. Coquasi Module Algebra over Hopf Coquasigroup

The following notion is different from the one in [15,16].

Definition 2.

Let H be a Hopf coquasigroup. Then,

(1) 

A unital algebra A is called a leftcoquasi H-module algebraif A is a left H-module such that, for all $a,b\in A$,

$$h·1_A=\epsilon \left(h\right)1_A,$$

(13)

and

$$h·\left(ab\right)=(h_{\left(1\right)}·a)(h_{\left(2\right)}·b);$$

(14)

(2) 

If A is a left coquasi H-module algebra, we can define the smash product $A\#H=A\otimes H$ with a multiplication given by

$$\begin{array}{c}\hfill (a\#h)(b\#l)=\sum a(h_{\left(1\right)}·b)\#h_{\left(2\right)}l\end{array}$$

(15)

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

Example 3.

(1) Let H be a Hopf coquasigroup. Then $\mathbb{K}$ is a left coquasi H-module algebra with the trivial action given by $h·a=\epsilon \left(h\right)a$, for $h\in H,a\in \mathbb{K}$. Thus, we have $\mathbb{K}\#H\cong H$;

(2) Let H be a finite-dimensional Hopf quasigroup. Then, $H^{*}$ be a finite-dimensional Hopf coquasigroup. A unital algebra H can be regarded as a left coquasi $H^{*}$-module algebra with ⇀. Hence, we can form smash product $H\#H^{*}$ with the following product:

$$(h\#h^{*})(l\#l^{*})=\sum ⟨h_{\left(1\right)}^{*},l_{\left(2\right)}⟩h{l}_{\left(1\right)}\#h_{\left(2\right)}^{*}{l}^{*}$$

(16)

for any $h,l\in H$ and $h^{*},l^{*}\in H^{*}$. In fact, for any $h,l\in H$ and $h^{*},l^{*}\in H^{*}$

$$\begin{array}{ccc}\hfill \left(h^{*}{l}^{*}\right)\rightharpoonup h& =& \sum h_{\left(1\right)}⟨h^{*}{l}^{*},h_{\left(2\right)}⟩\hfill \\ & =& \sum h_{\left(1\right)}⟨h^{*},h_{\left(2\right)\left(1\right)}⟩⟨l^{*},h_{\left(2\right)\left(2\right)}⟩\hfill \\ & =& \sum h_{\left(1\right)\left(1\right)}⟨h^{*},h_{\left(1\right)\left(2\right)}⟩⟨l^{*},h_{\left(2\right)}⟩\hfill \\ & =& \sum (h^{*}\rightharpoonup h_{\left(1\right)})⟨l^{*},h_{\left(2\right)}⟩\hfill \\ & =& h^{*}\rightharpoonup (l^{*}\rightharpoonup h)\hfill \end{array}$$

and so H is a left $H^{*}$-module. It is easy to see that $h^{*}\rightharpoonup 1_H=⟨h^{*},1_H⟩1_H$. By Proposition 2(c), we obtain Equation (6).

Proposition 4.

Let A be a left coquasi H-module algebra. If A is a unital associative algebra, then $A\#H$ is a unital associative algebra with unit $1_A\#1_H$ if and only if

$$\begin{array}{c}\hfill \sum h_{\left(1\right)}·\left(ab\right)\otimes h_{\left(2\right)}=\sum (h_{\left(1\right)}·a)(h_{\left(2\right)\left(1\right)}·b)\otimes h_{\left(2\right)\left(2\right)}.\end{array}$$

(17)

Proof. 

Obviously, $A\#H$ is a unital algebra with unit $1_A\#1_H$. Furthermore, for any $a,b,\in A$ and $x,y,z\in H$, we have

$$\begin{array}{ccc}\hfill \left[\right(a\#x\left)\right(b\#y\left)\right](c\#z)& =& \sum [a(x_{\left(1\right)}·b)\#x_{\left(2\right)}y](c\#z)\hfill \\ & =& \sum \left[a(x_{\left(1\right)}·b)\right][{\left(x_{\left(2\right)}y\right)}_{\left(1\right)}·c]\#{\left(x_{\left(2\right)}y\right)}_{\left(2\right)}z\hfill \\ & =& \sum \left[a(x_{\left(1\right)}·b)\right][x_{\left(2\right)\left(1\right)}{y}_{\left(1\right)}·c]\#\left[x_{\left(2\right)\left(2\right)}{y}_{\left(2\right)}\right]z\hfill \\ & =& \sum a\left[(x_{\left(1\right)}·b)(x_{\left(2\right)\left(1\right)}·(y_{\left(1\right)}·c))\right]\#x_{\left(2\right)\left(2\right)}\left[y_{\left(2\right)}z\right]\hfill \\ & \stackrel{\left(16\right)}{=}& \sum a[x_{\left(1\right)}·\left(b(y_{\left(1\right)}·c)\right)]\#x_{\left(2\right)}\left[y_{\left(2\right)}z\right]\hfill \\ & =& \sum (a\#x)[b(y_{\left(1\right)}·c)\#y_{\left(2\right)}z]\hfill \\ & =& (a\#x)\left[\right(b\#y\left)\right(c\#z\left)\right].\hfill \end{array}$$

Obviously, $A\#H=(A\#1_H)(1_A\#H)$. □

Remark 4.

(1) In [16], we replace Equation (13) with Equation (16) to define a left quasi H-module algebra;

(2) The unital algebra $H\#H^{*}$ in Example 3 is not associative.

Proposition 5.

Let H be a finite-dimensional Hopf quasigroup. If A is a left H-quasimodule algebra, then $A★H$ becomes a left coquasi $H^{*}$-module algebra via

$$h^{*}·(a★h)=a★(h^{*}\rightharpoonup h)$$

for any $h^{*}\in H^{*},a\in A$ and $h\in H$. Then, we have a smash product $(A★H)\#H^{*}$.

Proof. 

By the proof of Example 3, it is easy to obtain that $A★H$ is a left $H^{*}$-module.

We now prove Equations (13) and (14). For Equation (13), we have, for $h^{*}\in H^{*},$

$$h^{*}·(1_A★1_H)=1_A★(h^{*}\rightharpoonup 1_H)=⟨h^{*},1_H⟩1_A★1_H.$$

As for Equation (14), we compute for $h^{*}\in H^{*},a,b\in A$ and $h,l\in H$,

$$\begin{array}{ccc}& & h^{*}·\left[(a★h)(b★l)\right]\hfill \\ & =& \sum h^{*}·[a(h_{\left(1\right)}·b)★h_{\left(2\right)}l]\hfill \\ & =& \sum a(h_{\left(1\right)}·b)★h^{*}\rightharpoonup \left(h_{\left(2\right)}l\right)\hfill \\ & =& \sum a(h_{\left(1\right)}·b)★(h_{\left(1\right)}^{*}\rightharpoonup h_{\left(2\right)})(h_{\left(2\right)}^{*}\rightharpoonup l)\hfill \\ & =& \sum a(h_{\left(1\right)}·b)★h_{\left(2\right)}{l}_{\left(1\right)}⟨h_{\left(1\right)}^{*},h_{\left(3\right)}⟩⟨h_{\left(2\right)}^{*},l_{\left(2\right)}⟩\hfill \\ & =& \sum a(h_{\left(1\right)\left(1\right)}·b)★h_{\left(1\right)\left(2\right)}{l}_{\left(1\right)}⟨h_{\left(1\right)}^{*},h_{\left(2\right)}⟩⟨h_{\left(2\right)}^{*},l_{\left(2\right)}⟩\hfill \\ & =& \sum (a★h_{\left(1\right)})(b★l_{\left(1\right)})⟨h_{\left(1\right)}^{*},h_{\left(2\right)}⟩⟨h_{\left(2\right)}^{*},l_{\left(2\right)}⟩\hfill \\ & =& \sum [a★h^{*}\rightharpoonup h][b★h_{\left(2\right)\left(1\right)}^{*}\rightharpoonup l]\hfill \\ & =& \sum [h_{\left(1\right)}^{*}·(a★h)][h_{\left(2\right)}^{*}·(b★l)].\hfill \end{array}$$

Then, we have a smash product $(A★H)\#H^{*}$. □

4. Duality Theorem

In this section, let H be a finite-dimensional Hopf quasigroup and A a left H-quasimodule algebra. We will prove our duality theorem.

Lemma 1.

Let Q be a Hopf coquasigroup and B a left coquasi H-module algebra. Then, there is a unital homomorphism

$${\mathrm{\Lambda }}_{B,Q}:B\#Q⟶End\left(B\right)$$

(18)

given by ${\mathrm{\Lambda }}_{B,Q}(a\#h)\left(b\right)=a(h·b)$ for any $h\in Q$ and $a,b\in B$. Furthermore, if B is associative, then ${\mathrm{\Lambda }}_{B,Q}$ is an algebra homomorphism.

Proof. 

It is easy to see that ${\mathrm{\Lambda }}_{B,Q}(1_B\#1_Q)\left(1_B\right)=1_B$. If B is associative, then, for any $h,l\in Q$ and $a,b,c\in B$, we have

$$\begin{array}{ccc}\hfill {\mathrm{\Lambda }}_{B,Q}\left[(a\#h)(b\#l)\right]\left(c\right)& =& \sum {\mathrm{\Lambda }}_{B,Q}(a(h_{\left(1\right)}·b)\#h_{\left(2\right)}l)\left(c\right)\hfill \\ & =& \sum \left[(a\left(h_{\left(1\right)}\right)·b)\right][\left(h_{\left(2\right)}l\right)·c]\hfill \\ & =& \sum a[(h_{\left(1\right)}·b)(h_{\left(2\right)}·(l·c))]\hfill \\ & =& a[h·[b(l·c)\left]\right]\hfill \\ & =& [{\mathrm{\Lambda }}_{B,Q}(a\#h)\circ {\mathrm{\Lambda }}_{B,Q}(b\#l)]\left(c\right).\hfill \end{array}$$

□

Remark 5.

For Equation (9), in the special case where $B=H$, a Hopf quasigroup, and $Q=H^{*}$, the Hopf coquasigroup. By Example 3(2), we have $H\#H^{*}$. Then, there is a unital algebra anti-homomorphism

$${\mathrm{\Gamma }}_{H,H^{*}}:H^{*}⟶End\left(H\right)$$

(19)

given by ${\mathrm{\Gamma }}_{H,H^{*}}\left(h^{*}\right)\left(h\right)=h\leftharpoonup h^{*}$ for any $h\in H$ and $h^{*}\in H^{*}$.

Let H be a finite-dimensional Hopf quasigroup and A a left H-quasimodule algebra. Then, we have the smash product $A★H$. By using the map $\mathrm{\Lambda }$ from Equation (9) and Proposition 2, we define:

$$\mathrm{\Phi }={\mathrm{\Lambda }}_{A★H,}:(A★H)\#H^{*}⟶End(A★H)$$

and by the left regular representation ${\lambda }_l:AEnd\left(A\right),a\mapsto (x\mapsto ax)$, one defines

$$\mathrm{\Psi }={\lambda }_l\otimes {\mathrm{\Lambda }}_{H,H^{*}}:A\otimes (H\#H^{*})⟶End(A★H).$$

That is,

$$\mathrm{\Phi }((a★h)\#h^{*})(b★l)=(a★h)(b★(h^{*}\rightharpoonup l))$$

and

$$\mathrm{\Psi }(a\otimes (h\#h^{*}))(b★l)=ab★h(h^{*}\rightharpoonup l)$$

for any $a,b\in A,h,l\in H$ and $h^{*}\in H^{*}$.

Remark 6.

We notice that $id\otimes {\mathrm{\Gamma }}_{H,H^{*}}\left(h^{*}\right)=\mathrm{\Psi }(1\otimes v)$ for some $v\in H\#H^{*}$ for $h^{*}\in H^{*}$.

Lemma 2.

With notations above, Φ and Ψ do not preserve multiplication. However, we have

(i) 

$\mathrm{\Phi }((a★h)\#h^{*})=\mathrm{\Phi }((a★1)\#\epsilon )\circ \mathrm{\Phi }((1★h)\#h^{*});$

(ii) 

$\mathrm{\Psi }(a\otimes (h\#h^{*}))=\mathrm{\Psi }(a\otimes (1\#\epsilon ))\circ \mathrm{\Psi }(1\otimes (h\#h^{*}));$

(iii) 

$\mathrm{\Phi }((1★h)\#h^{*})=\mathrm{\Phi }((1★h)\#\epsilon )\circ \mathrm{\Phi }((1★1)\#h^{*});$

(iv) 

$\mathrm{\Psi }(1\otimes (h\#h^{*}))=\mathrm{\Psi }(1\otimes (h\#\epsilon ))\circ \mathrm{\Psi }(1\otimes (1\#h^{*})).$

Proof. 

For any $a,b\in A,h,l\in H$ and $h^{*}\in H^{*}$, we have

$$\begin{array}{ccc}& & \mathrm{\Phi }((a★h)\#h^{*})(b★l)\hfill \\ & =& (a★h)(b★(h^{*}\rightharpoonup l))\hfill \\ & =& a(h_{\left(1\right)}▹b)★h_{\left(2\right)}(h^{*}\rightharpoonup l)\hfill \\ & =& (a★1)[(h_{\left(1\right)}▹b)★h_{\left(2\right)}(h^{*}\rightharpoonup l)]\hfill \\ & =& \mathrm{\Phi }((a★1)\#\epsilon )[(h_{\left(1\right)}▹b)★h_{\left(2\right)}(h^{*}\rightharpoonup l)]\hfill \\ & =& \mathrm{\Phi }((a★1)\#\epsilon )(1★h)(b★h^{*}\rightharpoonup l)\hfill \\ & =& [\mathrm{\Phi }((a★1)\#\epsilon )\circ \mathrm{\Phi }((1★h)\#h^{*})](b★l).\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}& & \mathrm{\Psi }(a\otimes (h\#h^{*}))(b★l)\hfill \\ & =& ab★h(h^{*}\rightharpoonup l)\hfill \\ & =& \mathrm{\Psi }(a\otimes (1\#\epsilon ))(b\otimes h(h^{*}\rightharpoonup l))\hfill \\ & =& [\mathrm{\Psi }(a\otimes (1\#\epsilon ))\circ \mathrm{\Psi }(1\otimes (h\#h^{*}))](b★l).\hfill \end{array}$$

(iii) and (iv). Straightforward. □

Lemma 3.

With notations above, Φ and Ψ are injective linear maps.

Proof. 

In order to prove that $\mathrm{\Phi }$ and $\mathrm{\Psi }$ are injective maps, we consider the following injective linear maps ${\mathrm{\Phi }}^{\prime },{\mathrm{\Psi }}^{\prime }$ and $\mathrm{\Theta }$:

$$\begin{array}{ccc}& & {\mathrm{\Phi }}^{\prime }:(A★H)\#H^{*}⟶End(A★H),{\mathrm{\Phi }}^{\prime }((a★h)\#h^{*})(b★l)=⟨h^{*},l⟩(a★h)(b★1),\hfill \\ & & {\mathrm{\Psi }}^{\prime }:A\otimes (H\#H^{*})⟶End(A★H),{\mathrm{\Psi }}^{\prime }(a\otimes (h\#h^{*}))(b★l)=⟨h^{*},l⟩ab★h,\hfill \\ & & \mathrm{\Theta }:End(A★H)⟶End(A★H),\mathrm{\Theta }\left(f\right)(b★l)=\sum \left[f(b★l_{\left(2\right)})\right](1★l_{\left(1\right)})\hfill \end{array}$$

for any $a,b\in A,h,l\in H,h^{*}\in H^{*}$ and $f\in End(A★H)$.

Let $x\in Ker\left({\mathrm{\Phi }}^{\prime }\right)$ and write $x={\sum }_{i=1}^n{y}_i★h^{*}i$, where $y_i\in A★H$ and $\{h_1^{*},h_2^{*},\cdots ,h_n^{*}\}$ is a linearly independent subset of $H^{*}$. Choose $h_1,h_2,·h_n$ such that $h_i^{*}\left(h_j\right)={\delta }_{ij}$, with $1\le i,j,\le n$. Then, $0={\mathrm{\Phi }}^{\prime }\left(x\right)(1★k_i)=y_i$ for all i, so that $x=0$. Thus, ${\mathrm{\Phi }}^{\prime }$ is injective.

Similarly, we can prove that ${\mathrm{\Psi }}^{\prime }$ is injective.

To see that $\mathrm{\Theta }$ is injective (actually it is bijective), we construct a left inverse for $\mathrm{\Theta }$. We define $\mathrm{Y}:End(A★H)⟶End(A★H)$ by

$$\mathrm{Y}\left(f\right)(b★l)=\sum \left[f(b★l_{\left(2\right)})\right](1★S^{-1}\left(l_{\left(1\right)}\right)).$$

Then, we compute

$$\begin{array}{ccc}\hfill (\mathrm{Y}\circ \mathrm{\Theta })\left(f\right)(b★l)& =& \sum \left[\mathrm{\Theta }\left(f\right)(b★l_{\left(2\right)})\right](1★S^{-1}\left(l_{\left(1\right)}\right))\hfill \\ & =& \sum \left[\left[f(b★l_{\left(2\right)\left(2\right)})\right](1★l_{\left(2\right)\left(1\right)})\right](1★S^{-1}\left(l_{\left(1\right)}\right))\hfill \\ & =& \sum \left[\left[f(b★l_{\left(2\right)})\right](1★l_{\left(1\right)\left(2\right)})\right](1★S^{-1}\left(l_{\left(1\right)\left(1\right)}\right))\hfill \\ & =& \sum u_i★\left(v_i{l}_{\left(1\right)\left(2\right)}\right)S^{-1}\left(l_{\left(1\right)\left(1\right)}\right){\displaystyle )}\hfill \\ & \stackrel{\left(3\right)}{=}& \sum u_i★v_i\epsilon \left(l_{\left(1\right)}\right)\hfill \\ & =& \sum f(b★l_{\left(2\right)})\epsilon \left(l_{\left(1\right)}\right)\hfill \\ & =& f(b★l)\hfill \end{array}$$

where we write $f(b★l_{\left(2\right)})={\sum }_i{u}_i\otimes v_i$, use the Remark after Definition 1, and use the coassociativity in the Hopf quasigroup.

Similarly, we also have:

$$\begin{array}{ccc}\hfill (\mathrm{\Theta }\circ \mathrm{Y})\left(f\right)(b★l)& =& \sum \left[\mathrm{Y}\left(f\right)(b★l_{\left(2\right)})\right](1★l_{\left(1\right)})\hfill \\ & =& \sum {\displaystyle [}\left[f(b★l_{\left(2\right)\left(2\right)})\right](1★S^{-1}\left(l_{\left(2\right)\left(1\right)}\right)](1★l_{\left(1\right)})\hfill \\ & =& \sum {\displaystyle [}\left[f(b★l_{\left(2\right)})\right](1★S^{-1}\left(l_{\left(1\right)\left(2\right)}\right)](1★l_{\left(1\right)\left(1\right)})\hfill \\ & =& \sum u_i★\left(v_i{S}^{-1}\left(l_{\left(1\right)\left(2\right)}\right)\right)l_{\left(1\right)\left(1\right)}\hfill \\ & \stackrel{\left(3\right)}{=}& \sum u_i★v_i\epsilon \left(l_{\left(1\right)}\right)\hfill \\ & =& \sum f(b★l_{\left(2\right)})\epsilon \left(l_{\left(1\right)}\right)\hfill \\ & =& f(b★l).\hfill \end{array}$$

Therefore, $\mathrm{Y}$ is a two-sided inverse for $\mathrm{\Theta }$.

Next, we will show that $\mathrm{\Phi }=\mathrm{\Theta }\circ {\mathrm{\Phi }}^{\prime }$ and $\mathrm{\Psi }=\mathrm{\Theta }\circ {\mathrm{\Psi }}^{\prime }$. For the first one, we have

$$\begin{array}{ccc}\hfill (\mathrm{\Theta }\circ {\mathrm{\Phi }}^{\prime })((a★h)\#h^{*})(b★l)& =& \mathrm{\Theta }\left[{\mathrm{\Phi }}^{\prime }((a★h)\#h^{*})\right](b★l)\hfill \\ & =& \sum {\mathrm{\Phi }}^{\prime }((a★h)\#h^{*})(b★l_{\left(2\right)})(1★l_{\left(1\right)})\hfill \\ & =& \sum \left[(a★h)(b★1)\right](1★l_{\left(1\right)})⟨h^{*},l_{\left(2\right)}⟩\hfill \\ & =& \sum [a(h_{\left(1\right)}▹b)★h_{\left(2\right)}{l}_{\left(1\right)}]⟨h^{*},l_{\left(2\right)}⟩\hfill \\ & =& \sum (a★h)(b★l_{\left(1\right)})⟨h^{*},l_{\left(2\right)}⟩\hfill \\ & =& (a★h)(b★(h^{*}\rightharpoonup l))\hfill \\ & =& \mathrm{\Phi }((a★h)\#h^{*})(b★l),\hfill \end{array}$$

and, for the second one, we compute as follows:

$$\begin{array}{ccc}\hfill (\mathrm{\Theta }\circ {\mathrm{\Psi }}^{\prime })(a\otimes (h\#h^{*}))(b★l)& =& \mathrm{\Theta }\left[{\mathrm{\Psi }}^{\prime }(a\otimes (h\#h^{*}))\right](b★l)\hfill \\ & =& \sum \left[{\mathrm{\Psi }}^{\prime }(a\otimes (h\#h^{*}))(b★l_{\left(2\right)})\right](1★l_{\left(1\right)})\hfill \\ & =& \sum (ab★h)(1★l_{\left(1\right)})⟨h^{*},l_{\left(2\right)}⟩\hfill \\ & =& \sum (ab★h{l}_{\left(1\right)})⟨h^{*},l_{\left(2\right)}⟩\hfill \\ & =& ab★h(h^{*}\rightharpoonup l)\hfill \\ & =& \mathrm{\Psi }(a\otimes (h\#h^{*}))(b★l).\hfill \end{array}$$

This shows that $\mathrm{\Phi }$ and $\mathrm{\Psi }$ are injective linear maps. □

Corollary 1.

Let H be a finite-dimensional Hopf quasigroup of dimension $n<\infty$. Then, ${\mathrm{\Lambda }}_{H,H^{*}}$ is a bijective linear map, so that $H\#H^{*}\cong End\left(H\right)\cong M_n\left(\mathbb{K}\right)$, the algebra of $n\times n$ matrices over $\mathbb{K}$.

Proof. 

By Example 2(1), we have $\mathbb{K}★H\cong H$ and so $\mathrm{\Psi }={\mathrm{\Lambda }}_{H,H^{*}}$, so that ${\mathrm{\Lambda }}_{H,H^{*}}$ is injective. Observe that $dim(H\#H^{*})=n^2=dim\left(End\left(H\right)\right)$ and so ${\mathrm{\Lambda }}_{H,H^{*}}$ is a bijective linear map. □

We next define a map $\mathrm{\Xi }\in End(A★H)$ by

$$\mathrm{\Xi }(b★l)=\sum (S^{-1}\left(l_{\left(1\right)}\right)▹b)★l_{\left(2\right)}$$

for any $b\in A$ and $l\in H$.

Lemma 4.

With notations above, Ξ is invertible with inverse Ω given by

$$\mathrm{\Omega }(b★l)=\sum (l_{\left(1\right)}▹b)★l_{\left(2\right)}$$

for any $b\in A$ and $l\in H$.

Proof. 

For any $b\in A$ and $l\in H$, we have

$$\begin{array}{ccc}\hfill \mathrm{\Xi }\left(\mathrm{\Omega }\right(b★l\left)\right)& =& \sum \mathrm{\Xi }{\displaystyle [}(l_{\left(1\right)}▹b)★l_{\left(2\right)}{\displaystyle ]}\hfill \\ & =& \sum (S^{-1}\left(l_{\left(2\right)\left(1\right)}\right)▹(l_{\left(1\right)}▹b))★l_{\left(2\right)\left(2\right)}\hfill \\ & =& \sum (S^{-1}\left(l_{\left(1\right)\left(2\right)}\right)▹(l_{\left(1\right)\left(1\right)}▹b))★l_{\left(2\right)}\hfill \\ & =& \sum \epsilon \left(l_{\left(1\right)}\right)b★l_{\left(2\right)}\hfill \\ & =& \sum b★l,\hfill \end{array}$$

and so $\mathrm{\Xi }\circ \mathrm{\Omega }=id$. Meanwhile, we have

$$\begin{array}{ccc}\hfill \mathrm{\Omega }\left(\mathrm{\Xi }\right(b★l\left)\right)& =& \sum \mathrm{\Omega }{\displaystyle [}(S^{-1}\left(l_{\left(1\right)}\right)▹b)★l_{\left(2\right)}{\displaystyle ]}\hfill \\ & =& \sum (l_{\left(2\right)\left(1\right)}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b))★l_{\left(2\right)\left(2\right)}\hfill \\ & =& \sum (l_{\left(1\right)\left(2\right)}▹(S^{-1}\left(l_{\left(1\right)\left(1\right)}\right)▹b))★l_{\left(2\right)}\hfill \\ & =& \sum \epsilon \left(l_{\left(1\right)}\right)b★l_{\left(2\right)}\hfill \\ & =& \sum b★l,\hfill \end{array}$$

and thus $\mathrm{\Omega }\circ \mathrm{\Xi }=id$. □

Let H be a finite-dimensional Hopf quasigroup of dimension $n<\infty$. Let $\{h_1^{*},h_2^{*},\cdots ,h_n^{*}\}$ be a basis of $H^{*}$ and $\{h_1,h_2,\cdots ,h_n\}$ be its dual basis for H, i.e., so that $h_i^{*}\left(h_j\right)={\delta }_{ij}$, with $1\le i,j,\le n$.

Remark 7.

With notations above, let $a\in A$ and $a_i=h_i▹a$ with $i\in \{1,2,\cdots ,n\}$. For any $h\in H$, we let $h={\sum }_i{k}_i{h}_i$ with $k_i\in \mathbb{K}$. Then, $h▹a={\sum }_i{k}_i(h_i▹a)={\sum }_i{k}_i{a}_i$. However, $⟨h_j^{*},h⟩=⟨h_j^{*},{\sum }_i{k}_i{h}_i⟩=k_j$, and so

$$h▹a=\sum _i⟨h_i^{*},h⟩a_i.$$

(20)

Lemma 5.

With the notations above, fix $a\in A$ so that $a_i=h_i▹a$ with $i\in \{1,2,\cdots ,n\}$ and, for any $b\in A,h,l\in H$ and $h^{*}\in H^{*}$, we have the following identities:

(i) 

$\mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (1\#h^{*}))\circ \mathrm{\Xi }=\mathrm{\Phi }((1★1)\#h^{*});$

(ii) 

$\mathrm{\Omega }\circ \mathrm{\Psi }(a\otimes (1\#\epsilon ))\circ \mathrm{\Xi }={\sum }_{i=1}^n\mathrm{\Phi }((a_i★1)\#\epsilon )\circ \mathrm{\Omega }\circ (id\otimes {\mathrm{\Gamma }}_{H,H^{*}}\left(h_i^{*}\right))\circ \mathrm{\Xi };$

(iii) 

$\mathrm{\Xi }\circ \mathrm{\Phi }((a★1)\#\epsilon )\circ \mathrm{\Omega }={\sum }_{i=1}^n\mathrm{\Psi }((a_i\otimes (1\#\epsilon ))\circ (id\otimes {\mathrm{\Gamma }}_{H,H^{*}}\left(S^{*-1}\left(h_i^{*}\right)\right);$

(iv) 

$[\mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (h\#\epsilon ))\circ \mathrm{\Xi }](b★l)=\mathrm{\Phi }((1★h{l}_{\left(2\right)})\#\epsilon )(S^{-1}(l_{\left(1\right)}▹b)★1).$

Proof. 

For any $b\in A$ and $l\in H$.

(i) We check as follows:

$$\begin{array}{ccc}& & [\mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (1\#h^{*}))\circ \mathrm{\Xi }](b★l)\hfill \\ & =& \sum [\mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (1\#h^{*}))](S^{-1}\left(l_{\left(1\right)}\right)▹b)★l_{\left(2\right)}\hfill \\ & =& \sum \mathrm{\Omega }{\displaystyle [}(S^{-1}\left(l_{\left(1\right)}\right)▹b)★(h^{*}\rightharpoonup l_{\left(2\right)}){\displaystyle ]}\hfill \\ & =& \sum {\displaystyle [}{\left[(h^{*}\rightharpoonup l_{\left(2\right)})\right]}_{\left(1\right)}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b){\displaystyle ]}★{\left[(h^{*}\rightharpoonup l_{\left(2\right)})\right]}_{\left(2\right)}\hfill \\ & =& \sum {\displaystyle [}{(h^{*}\rightharpoonup l_{\left(2\right)})}_{\left(1\right)}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b){\displaystyle ]}★{\displaystyle [}{(h^{*}\rightharpoonup l_{\left(2\right)})}_{\left(2\right)}{\displaystyle ]}\hfill \\ & =& \sum [l_{\left(2\right)\left(1\right)}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b)]★(h^{*}\rightharpoonup l_{\left(2\right)\left(2\right)})\mathrm{b}\mathrm{y} \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} 1.3\left(\mathrm{a}\right)\hfill \\ & =& \sum [l_{\left(1\right)\left(2\right)}▹(S^{-1}\left(l_{\left(1\right)\left(1\right)}\right)▹b)]★(h^{*}\rightharpoonup l_{\left(2\right)})\hfill \\ & =& \sum b★(h^{*}\rightharpoonup l)\hfill \\ & =& \mathrm{\Phi }((1★1)\#h^{*})(b★l).\hfill \end{array}$$

For (ii), we have

$$\begin{array}{ccc}& & [\mathrm{\Omega }\circ \mathrm{\Psi }(a\otimes (1\#\epsilon ))\circ \mathrm{\Xi }](b★l)\hfill \\ & =& \sum [\mathrm{\Omega }\circ \mathrm{\Psi }(a\otimes (1\#\epsilon ))](S^{-1}\left(l_{\left(1\right)}\right)▹b)★l_{\left(2\right)}\hfill \\ & =& \sum \mathrm{\Omega }[a(S^{-1}\left(l_{\left(1\right)}\right)▹b)★(\epsilon \rightharpoonup l_{\left(2\right)})]\hfill \\ & =& \sum \mathrm{\Omega }[a(S^{-1}\left(l_{\left(1\right)}\right)▹b)★l_{\left(2\right)}]\hfill \\ & =& \sum \left[l_{\left(2\right)\left(1\right)}\left[a(S^{-1}\left(l_{\left(1\right)}\right)▹b)\right]\right]★l_{\left(2\right)\left(2\right)}\hfill \\ & =& \sum [l_{\left(2\right)\left(1\right)\left(1\right)}▹a][l_{\left(2\right)\left(1\right)\left(2\right)}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b)]★l_{\left(2\right)\left(2\right)}\hfill \\ & =& \sum [l_{\left(2\right)}▹a][l_{\left(3\right)}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b)]★l_{\left(4\right)}\hfill \\ & \stackrel{\left(11\right)}{=}& \sum \sum _{i=1}^n⟨h_i^{*},l_{\left(2\right)}⟩a_i[l_{\left(3\right)}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b)]★l_{\left(4\right)}\hfill \\ & =& \sum \sum _{i=1}^n(a_i★1)[l_{\left(3\right)}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b)★l_{\left(4\right)}]⟨h_i^{*},l_{\left(2\right)}⟩\hfill \\ & =& \sum \sum _{i=1}^n\left[\mathrm{\Phi }((a_i★1)\#\epsilon )\right][l_{\left(3\right)}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b)★l_{\left(4\right)}]⟨h_i^{*},l_{\left(2\right)}⟩\hfill \\ & =& \sum \sum _{i=1}^n\left[\mathrm{\Phi }((a_i★1)\#\epsilon )\right][l_{\left(2\right)}\leftharpoonup h_i^{*}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b)★l_{\left(3\right)}]\hfill \\ & =& \sum \sum _{i=1}^n\left[\mathrm{\Phi }((a_i★1)\#\epsilon )\right]\left[\right({(l_{\left(2\right)}\leftharpoonup h_i^{*})}_{\left(1\right)}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b)★({(l_{\left(2\right)}\leftharpoonup h_i^{*})}_{\left(2\right)}]\hfill \\ & & \mathrm{b}\mathrm{y} \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} 2\left(\mathrm{b}\right)\hfill \\ & =& \sum \sum _{i=1}^n\left[\mathrm{\Phi }((a_i★1)\#\epsilon )\right]\mathrm{\Omega }[(S^{-1}\left(l_{\left(1\right)}\right)▹b)★(l_{\left(2\right)}\leftharpoonup h_i^{*}]\hfill \\ & =& \sum \sum _{i=1}^n[\mathrm{\Phi }((a_i★1)\#\epsilon )\circ \mathrm{\Omega }][(S^{-1}\left(l_{\left(1\right)}\right)▹b)★(l_{\left(2\right)}\leftharpoonup h_i^{*}]\hfill \\ & =& \sum \sum _{i=1}^n[\mathrm{\Phi }((a_i★1)\#\epsilon )\circ \mathrm{\Omega }\circ (id\otimes {\mathrm{\Gamma }}_{H,H^{*}}\left(h_i^{*}\right))](S^{-1}\left(l_{\left(1\right)}\right)▹b)★l_{\left(2\right)}\hfill \\ & =& \sum _{i=1}^n[\mathrm{\Phi }((a_i★1)\#\epsilon )\circ \mathrm{\Omega }\circ (id\otimes {\mathrm{\Gamma }}_{H,H^{*}}\left(h_i^{*}\right))\circ \mathrm{\Xi }](b★l).\hfill \end{array}$$

For (iii), one has

$$\begin{array}{ccc}& & [\mathrm{\Xi }\circ \mathrm{\Phi }((a★1)\#\epsilon )\circ \mathrm{\Omega }](b★l)\hfill \\ & =& \sum [\mathrm{\Xi }\circ \mathrm{\Phi }((a★1)\#\epsilon )]((l_{\left(1\right)}▹b)★l_{\left(2\right)})\hfill \\ & =& \sum \mathrm{\Xi }\left[(a★1)((l_{\left(1\right)}▹b)★l_{\left(2\right)})\right]\hfill \\ & =& \sum \mathrm{\Xi }[a(l_{\left(1\right)}▹b)★l_{\left(2\right)}]\hfill \\ & =& \sum S^{-1}\left(l_{\left(2\right)\left(1\right)}\right)▹\left[a(l_{\left(1\right)}▹b)\right]★l_{\left(2\right)\left(2\right)}]\hfill \\ & =& \sum S^{-1}\left(l_{\left(2\right)}\right)▹\left[a(l_{\left(1\right)}▹b)\right]★l_{\left(3\right)}]\hfill \\ & =& \sum (S^{-1}\left(l_{\left(3\right)}\right)▹a)[S^{-1}\left(l_{\left(2\right)}\right)▹(l_{\left(1\right)}▹b)]★l_{\left(4\right)}]\hfill \\ & =& \sum (S^{-1}\left(l_{\left(1\right)}\right)▹a)b★l_{\left(2\right)}\hfill \\ & \stackrel{\left(11\right)}{=}& \sum _{i=1}^n⟨h_i^{*},S^{-1}\left(l_{\left(1\right)}\right)⟩a_{i}b★l_{\left(2\right)}\hfill \\ & =& \sum _{i=1}^n{a}_{i}b★l_{\left(2\right)}⟨S^{*-1}\left(h_i^{*}\right),l_{\left(1\right)}⟩\hfill \\ & =& \sum _{i=1}^n{a}_{i}b★(l\leftharpoonup S^{*-1}\left(h_i^{*}\right))\hfill \\ & =& \sum _{i=1}^n[\mathrm{\Psi }\left((a_i\otimes (1\#\epsilon ))\right](b★(l\leftharpoonup S^{*-1}\left(h_i^{*}\right)))\hfill \\ & =& \sum _{i=1}^n\left[\mathrm{\Psi }\right((a_i\otimes (1\#\epsilon ))\circ (id\otimes {\mathrm{\Gamma }}_{H,H^{*}}(S^{*-1}(h_i^{*}))](b★l).\hfill \end{array}$$

In addition, finally, for (iv), we have

$$\begin{array}{ccc}& & [\mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (h\#\epsilon ))\circ \mathrm{\Xi }](b★l)\hfill \\ & =& \sum [\mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (h\#\epsilon ))](S^{-1}\left(l_{\left(1\right)}\right)▹b)★l_{\left(2\right)}\hfill \\ & =& \sum \mathrm{\Omega }[(S^{-1}\left(l_{\left(1\right)}\right)▹b)★h{l}_{\left(2\right)}]\hfill \\ & =& \sum [{\left(h{l}_{\left(2\right)}\right)}_{\left(1\right)}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b)]★{\left(h{l}_{\left(2\right)}\right)}_{\left(2\right)}\hfill \\ & =& \sum [h_{\left(1\right)}{l}_{\left(2\right)}▹(S^{-1}\left(l_{\left(1\right)}\right)▹b)]★\left(h_{\left(2\right)}{l}_{\left(3\right)}\right)\hfill \\ & =& \sum [{\left(h_{\left(2\right)}l\right)}_{\left(2\right)}▹(S^{-1}({\left(h_{\left(2\right)}l\right)}_{\left(1\right)})h_{\left(1\right)}▹b)]★{\left(h_{\left(2\right)}l\right)}_{\left(3\right)}\hfill \\ & =& \mathrm{\Phi }((1★{\left(h_{\left(2\right)}l\right)}_{\left(2\right)})\#\epsilon )((S^{-1}({\left(h_{\left(2\right)}l\right)}_{\left(1\right)})h_{\left(1\right)}▹b)★1)\hfill \\ & =& \mathrm{\Phi }((1★h{l}_{\left(2\right)})\#\epsilon )(S^{-1}(l_{\left(1\right)}▹b)★1).\hfill \end{array}$$

This completes the proof. □

Remark 8.

In general, for any $h\in H$ and $h^{*}\in H^{*}$, we have

$$\mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (h\#h^{*}))\circ \mathrm{\Xi }\ne \mathrm{\Phi }((1★h)\#h^{*}).$$

Lemma 6.

Let B be a semigroup with a multiplication. Let $H,J\subseteq B$ be non-empty subsets of B. If there is an invertible element $\chi \in B$ so that $H={\chi }^{-1}J\chi$, then there exists a bijective map $\xi :H⟶J$ that preserves multiplication.

Proof. 

Define a map

$$\xi :H⟶J,h\mapsto \chi h{\chi }^{-1}.$$

Obviously, we can show that $\xi$ is bijective and $\xi \left(hl\right)=\chi \left(hl\right){\chi }^{-1}=\chi h{\chi }^{-1}\chi l{\chi }^{-1}=\xi \left(h\right)\xi \left(l\right)$ for $h,l\in H$. □

We are now in a position to prove the main theorem of this paper.

Theorem 1.

Let H be a finite-dimensional Hopf quasigroup with bijective antipode and A a left H-quasimodule algebra. Then,

$$(A★H)\#H^{*}\cong A\otimes (H\#H^{*}).$$

Proof. 

Let $a\in A,h\in H$ and $h^{*}\in H^{*}$. Firstly, we show that $\mathrm{\Omega }\circ \mathrm{\Psi }(a\otimes (h\#h^{*}))\circ \mathrm{\Xi }$ belong to $\mathrm{\Phi }((A★H)\#H^{*})$.

By the fact that $a\otimes (h\#h^{*})=(a\otimes (1\#\epsilon ))(1\otimes (h\#h^{*}))$ and Lemma 2, we have

$$\begin{array}{ccc}& & \mathrm{\Omega }\circ \mathrm{\Psi }(a\otimes (h\#h^{*}))\circ \mathrm{\Xi }\hfill \\ & =& \mathrm{\Omega }\circ \mathrm{\Psi }(a\otimes (1\#\epsilon ))\circ \mathrm{\Psi }(1\otimes (h\#h^{*}))\circ \mathrm{\Xi }\hfill \\ & =& [\mathrm{\Omega }\circ \mathrm{\Psi }(a\otimes (1\#\epsilon ))\circ \mathrm{\Xi }]\circ [\mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (h\#h^{*}))\circ \mathrm{\Xi }]\mathrm{b}\mathrm{y} \mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a} 4.\hfill \end{array}$$

Thus, it suffices to show that $\mathrm{\Omega }\circ \mathrm{\Psi }(a\otimes (1\#\epsilon \left)\right)\circ \mathrm{\Xi }$ and $\mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (h\#h^{*}))\circ \mathrm{\Xi }$ each belong to $\mathrm{\Phi }((A★H)\#H^{*})$. The first does by Lemma 5(ii). We also have

$$\begin{array}{ccc}\hfill \mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (h\#h^{*}))\circ \mathrm{\Xi }& =& \mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (h\#\epsilon ))\circ \mathrm{\Psi }(1\otimes (1\#h^{*}))\circ \mathrm{\Xi }\hfill \\ & =& [\mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (h\#\epsilon ))\circ \mathrm{\Xi }]\circ [\mathrm{\Omega }\circ \mathrm{\Psi }(1\otimes (1\#h^{*}))\circ \mathrm{\Xi }]\hfill \\ & \in & \mathrm{\Phi }((A★H)\#H^{*})\mathrm{b}\mathrm{y} \mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a} 5\left(\mathrm{i}\right)\left(\mathrm{i}v\mathrm{\right)}\hfill \end{array}$$

which implies that the second does also.

Then, we prove similarly that $\mathrm{\Xi }\circ \mathrm{\Phi }((a★h)\#h^{*})\circ \mathrm{\Omega }$ belongs to $\mathrm{\Psi }(A\otimes (H\#H^{*}))$. Actually, it follows from Lemma 5(i)(iii)(iv).

We now obtain

$$\mathrm{\Phi }((A★H)\#H^{*})={\mathrm{\Xi }}^{-1}\circ \mathrm{\Psi }(A\otimes (H\#H^{*}))\circ \mathrm{\Xi }.$$

By Lemma 6, our theorem is proved. □

Corollary 2.

Let H be a finite-dimensional Hopf quasigroup with bijective antipode and A a left H-quasimodule algebra. Then,

$$(A★H)\#H^{*}\cong A\otimes (H\#H^{*})\cong A\otimes M_n\left(\mathbb{K}\right)\cong M_n\left(A\right).$$

Proof. 

It follows Theorem 1 and Corollary 1. □

Example 4.

Let Q be a quasigroup (see [17]). Then, it follows from [1] (Proposition 4.7) that $H=\mathbb{K}Q$ is a Hopf quasigroup with a linear extension of the product and $\mathrm{\Delta }\left(h\right)=h\otimes h,\epsilon \left(h\right)=1$ and $S\left(h\right)=h^{-1}$ on the basis elements $h\in Q$.

If Q is a finite quasigroup, then ${\left(\mathbb{K}Q\right)}^{*}$ is a Hopf coquasigroup (see [1]). Explicitly, a basis of ${\left(\mathbb{K}Q\right)}^{*}$ is the set of projections $\{p_g\mid g\in Q\}$; that is, for any $g\in Q$ and $x={\sum }_{h\in Q}{\alpha }_{h}h\in \mathbb{K}Q$, $p_g\left(x\right)={\alpha }_g\in \mathbb{K}$. The set $\left\{p_g\right\}$ consists of orthogonal idempotents whose sum is 1. The comultiplication on ${\left(\mathbb{K}Q\right)}^{*}$ is given by $\mathrm{\Delta }\left(p_g\right)={\sum }_{h\in Q}{p}_{g{h}^{-1}}\otimes p_h$, and the counit is given by $\epsilon \left(p_g\right)={\delta }_{1,g}$ (where $\delta$ denotes the Kronecker delta).

Let A be a left $\mathbb{K}Q$-quasimodule algebra. Then, we have $1▹a=a$, $h▹\left(ab\right)=(h▹a)(h▹b)$, and

$$h▹(h^{-1}▹a)=h^{-1}▹(h▹a)=a$$

for all $h\in Q$ and $a,b\in A$.

We remark here that Q does not act as automorphism of A like a group acting as automorphism of A. In case of group G, we know that A is a Hopf algebra $\mathbb{K}G$-module algebra if and only if G acts as automorphism of A, and the smash product $A\#\mathbb{K}G=A\ast G$ is just the skew group ring of G over A (see [18]).

In our case of quasigroup Q, we have a skew quasigroup ring$A★Q$ of Q over A with a product:

$$\begin{array}{c}\hfill (a★x)(b★y)=a(x▹b)★xy\end{array}$$

(21)

for any $a,b\in A$ and $x,y\in Q$. We note that a skew quasigroup ring generally is not associative unless $\left(xy\right)▹a=x▹(y▹a)$ and Q is a group.

We know that ${\left(\mathbb{K}Q\right)}^{*}$ is a Hopf coquasigroup. Then, a unital algebra A is a left coquasi ${\left(\mathbb{K}Q\right)}^{*}$-module algebra if A is a left $\mathbb{K}Q$-module, i.e., Q-action, such that $p_g·1={\delta }_{1,g}1$ and

$$p_g·\left(ab\right)=\sum _{h\in Q}(p_{g{h}^{-1}}·a)(p_h·b)$$

for all $a,b\in A$. Then, we have a smash product $A\ast Q$ with a multiplication given by

$$(a\ast p_g)(b\ast p_l)=a(p_{g{l}^{-1}}·b)\ast p_l$$

where $a,b\in A$ and $g,l\in Q$.

In particular, by Example 3(2), when Q is finite, we have the smash product $Q\ast Q$ with the following product:

$$(h\ast p_g)(q\ast p_l)=hq\ast p_{q^{-1}g}$$

for any $h,g,q,l\in Q$.

By Corollary 2, we have

$$(A★Q)\ast Q\cong A\otimes (Q\ast Q)\cong A\otimes M_n\left(\mathbb{K}\right)\cong M_n\left(A\right).$$

In the end of this paper, we remark here that, when we consider a finite field ${\mathbb{Z}}_p$ (Galois field) with a prime p as a finite-dimensional Hopf quasigroup over ${\mathbb{Z}}_p$, we have a Hopf algebra ${\mathbb{Z}}_p$ with the coproduct $\mathrm{\Delta }\left(\right[a\left]\right)=\left[a\right]\left(\right[1]\otimes [1\left]\right)$ and the counit $\epsilon \left(\right[a\left]\right)=\left[a\right]$ for any $\left[a\right]\in {\mathbb{Z}}_p$. Then, we have

$$({\mathbb{Z}}_p★{\mathbb{Z}}_p)\#{\left({\mathbb{Z}}_p\right)}^{*}\cong ({\mathbb{Z}}_p\#{\left({\mathbb{Z}}_p\right)}^{*})\cong M_p\left({\mathbb{Z}}_p\right)$$

where we use the adjoint action of ${\mathbb{Z}}_p$ in the smash product ${\mathbb{Z}}_p★{\mathbb{Z}}_p$.

5. Conclusions and Further Research

As we mentioned already in the Introduction, Blattner and Montgomery obtained in [13] the duality theorem in the setting of Hopf algebras. In particular, if H is a finite-dimensional Hopf algebra and A is a left H-module algebra, then the duality theorem takes the form: $(A\#H)\#H^{*}\cong A\otimes (H\#H^{*})\cong M_n\left(A\right)$.

The dual space of a finite-dimensional Hopf algebra is a Hopf algebra. This duality breaks down for Hopf quasigroups, since the dual coalgebra of the algebra of a Hopf quasigroup is no longer a co-associative. This means that the dual space of a finite-dimensional Hopf quasigroup is not a Hopf quasigroup, but a Hopf coquasigroup, which are generalizations of Hopf algebras.

In this paper, we have studied two kinds of smash products “★” and “#” on the tensor product space $A\otimes H$ associated with a finite-dimensional Hopf quasigroup H and a left H-quasimodule algebra A. We have obtained an analogue of Blattner and Montgomery’s duality theorem in the general finite-dimensional Hopf quasigroup case in Section 4: $(A★H)\#H^{*}\cong A\otimes (H\#H^{*})\cong A\otimes M_n\left(\mathbb{K}\right)\cong M_n\left(A\right)$ (see Theorem 1 and Corollary 2). In addition, we have paid special attention to the finite quasigroup case (see Example 4). It is still not clear if the nicer results, obtained in the finite case, can be pushed forward to the infinite case so that better results can also be shown there. We expect, however, that this will not be easy, neither to prove these results if they are true nor to find counter examples if they are not.

Finally, constructing an analogue of Blattner and Montgomery’s duality theorem in the general finite-dimensional Hopf coquasigroup is not so easy to do.