Multiplier Left Hopf Algebras

Abstract

In this paper, we introduce and study the notion of a multiplier left Hopf algebra, which can be seen as an extension of the Van Daele’s multiplier Hopf algebras and the Green–Nichols–Taft’s left Hopf algebras. In particular, we investigate the relation between the notion of the Van Daele’s left multiplier Hopf algebras and the one of our multiplier left Hopf algebras. Finally, we determine the case when a multiplier left Hopf algebra becomes a multiplier Hopf algebra.

1. Introduction

There is given a bialgebra A over a field $\mathbf{k}$. That means that A is an associative algebra with a product $m:A\otimes A⟶A$, an identity $\mu :\mathbf{k}⟶A$, a coassociative coproduct $\mathsf{\Delta }:A⟶A\otimes A$, and a counit $\epsilon :A⟶\mathbf{k}$ so that $\mathsf{\Delta }$ and $\epsilon$ are algebra homomorphisms. The set $End\left(A\right)$ of all $\mathbf{k}$-linear maps of A forms a monoid regarding to the convolution product $f\ast g=m(f\otimes g)\mathsf{\Delta }$ and the identity $\mu \epsilon$.

A bialgebra H is called a left Hopf algebra [1] if there exists an S in $End\left(H\right)$ satisfying $S\ast id=\mu \epsilon$, i.e., if $\mathsf{\Delta }\left(h\right)=\sum h_{\left(1\right)}\otimes h_{\left(2\right)}$, then $(\forall h\in H):\sum S\left(h_{\left(1\right)}\right)h_{\left(2\right)}=\epsilon \left(h\right)1$; this map S is termed as a left antipode of H. If S also satisfies $S\ast id=id\ast S=\mu \epsilon$ in $End\left(H\right)$, i.e., $\sum h_{\left(1\right)}S\left(h_{\left(2\right)}\right)=\epsilon \left(h\right)1$ also holds, then the bialgebra H is called a Hopf algebra. This map S is unique; moreover, it is an algebra and coalgebra antimorphism of H (cf, [2]).

For a Hopf algebra A, if we drop the assumption that A has an identity and if we allow the product $\mathsf{\Delta }$ to have values in the so-called multiplier algebra $M(A\otimes A)$, we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra (see [3,4]). This theory has been recently developed to the new one of weak multiplier Hopf algebras (see [5,6,7,8]).

The purpose of the present paper is to consider the notion of a left Hopf algebra introduced by Green, Nichols, and Taft [1] in the context of multiplier Hopf algebras as defined by Van Daele [3].

This article is laid out as follows: The main part of Section 2 gives some necessary definitions, notations, and some results about left multiplier Hopf algebras. Section 3 mainly gives a study of the bijective and anti-bialgebra homomorphic properties of left antipodes on a left Hopf algebra (see Theorems 1 and 2).

In Section 4, we introduce and study the notion of a multiplier left (right) Hopf algebra (Definition 3); we mainly investigate construction and properties of the counit (see Theorem 3) and the left antipode’s properties (see Theorem 4). In Section 5, we discuss the relation between multiplier Hopf algebras and multiplier left Hopf algebras (see Theorem 6) and in Section 6, we give an explicit example.

2. Preliminaries

Unless otherwise specified, all vector spaces, linear maps, and tensor products are considered over an algebraically closed field $\mathbf{k}$ of characteristic 0. In this section, we present some facts and notations of (left) Hopf algebras that will be used throughout this paper and can be found in [1,2].

Let A be an associative algebra for which we do not require an identity, but we assume that the multiplication is non-degenerate. For this A, one has the multiplier algebra $M\left(A\right)$. Obviously, $A\subseteq M\left(A\right)$, in particular, when A has an identity, one has $M\left(A\right)=A$; $M\left(A\right)$ can be characterized as the largest unital algebra as an essential ideal. We denote the multiplier of $A\otimes A$ by $M(A\otimes A)$. The following conclusions is very natural:

$$A\otimes A\subseteq M\left(A\right)\otimes M\left(A\right)\subseteq M(A\otimes A).$$

Let $\mathsf{\Delta }:A⟶M(A\otimes A)$ be a linear map, then have the following four Galois linear maps from $A\otimes A$ to $M(A\otimes A)$ given by (see [3,4,9,10])

$$\begin{array}{ccc}& & T^{lr}(a\otimes b)=\mathsf{\Delta }\left(a\right)(1\otimes b)\mathrm{and}{T}^{rl}(a\otimes b)=(a\otimes 1)\mathsf{\Delta }\left(b\right),\hfill \\ & & T^{rr}(a\otimes b)=(1\otimes b)\mathsf{\Delta }\left(a\right)\mathrm{and}{T}^{ll}(a\otimes b)=\mathsf{\Delta }\left(b\right)(a\otimes 1).\hfill \end{array}$$

These linear maps are called the canonical maps related to $\mathsf{\Delta }$. These maps are called regular if their images are in $A\otimes A$. The coproduct $\mathsf{\Delta }$ is called coassociative if

$$\left(\right(\mathsf{\Delta }\otimes \iota \left)\right(\mathsf{\Delta }\left(a\right)(1\otimes b)\left)\right)(c\otimes 1\otimes 1)=\left(\right(\iota \otimes \mathsf{\Delta }\left)\right(\mathsf{\Delta }\left(a\right)(c\otimes 1)\left)\right)(1\otimes 1\otimes b)$$

for all $a,b,c\in A$.

We assume A is an associative algebra with a non-degenerate product and with or without an identity.

A left multiplier Hopf algebra is a pair $(A,\mathsf{\Delta })$; here, $\mathsf{\Delta }$ is coassociative and an algebra map so that $T^{lr}$ and $T^{ll}$ are not only regular but also bijections of $A\otimes A$ (see Definition 1.5 in [9]).

In the same way, we can define a right multiplier Hopf algebra by the maps $T^{rl}$ and $T^{rr}$.

3. About Antipodes of Left Hopf Algebras

The objective of this section is to study the bijective and anti-bialgebra homomorphic properties of left antipodes about a left Hopf algebra.

As stated in the introduction, a left Hopf algebra H is a unital bialgebra $(H,\mathsf{\Delta },\epsilon )$ with an identity $1_H$, a coproduct $\mathsf{\Delta }$, a counit $\epsilon$, and a left antipode S. We will use the standard Sweedler’s notation (cf. [2]): $\mathsf{\Delta }\left(h\right)=\sum h_{\left(1\right)}\otimes h_{\left(2\right)}$. The axioms of the left antipode and the counit can be written as follows:

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

for any $h\in H$.

Similarly, A right Hopf algebra H is a bialgebra $(H,\mathsf{\Delta },\epsilon )$ with a linear map S satisfying the following identity:

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

for any $h\in H$.

Remark 1. 

(1) 

Let C be the coalgebras of $n\times n$ comatrices for $n\ge 2$. Then, there are the free left Hopf algebras on C. This is the first example of left Hopf algebras given in [1]. These are not Hopf algebras.

The specific left antipode constructed was both an algebra and a coalgebra antimorphism.

(2) 

For a left Hopf algebra with a left antipode S, if S also obeys the formula $id\ast S=\mu \epsilon$, then there are infinite number of left antipodes on H (see Theorem 3 in [11]).

(3) 

In the case of a left Hopf algebra or a right Hopf algebra, the antipode S is not generally an algebra anti-homomorphism and a coalgebra anti-homomorphism.

The following result is the motivation for our definitions of multiplier left (right) Hopf algebras.

Proposition 1. 

Let H be a unital bialgebra with a linear map $S:H⟶H$. We consider the following two linear maps from $H\otimes H$ to $H\otimes H$ defined by

$$\begin{array}{c}\hfill R^{lr}(a\otimes b)=\left((1\otimes S)\mathsf{\Delta }\left(a\right)\right)(1\otimes b)and{R}^{rl}(a\otimes b)=(a\otimes 1)\left((S\otimes 1)\mathsf{\Delta }\left(b\right)\right)\end{array}$$

for any $a,b\in H$.

(1) 

If H is a left Hopf algebra, then we have

(i) 

$R^{lr}\circ T^{lr}=i{d}_{A\otimes A}$ and $T^{lr}\circ R^{lr}\ne i{d}_{A\otimes A}$.

(ii) 

$T^{rl}\circ R^{rl}=i{d}_{A\otimes A}$ and $R^{rl}\circ T^{rl}\ne i{d}_{A\otimes A}$.

(iii) 

$\sum T^{rl}(cS\left(a_{\left(1\right)}{b}_{\left(1\right)}\right)\otimes a_{\left(2\right)}{b}_{\left(2\right)})=\sum T^{rl}\left(cS\left(b_{\left(1\right)}\right)S\left(a_{\left(1\right)}\right)\right)\otimes a_{\left(2\right)}{b}_{\left(2\right)})$, for any $a,b,c\in H$.

(2) 

If H is a right Hopf algebra, then we have

(i) 

$R^{lr}\circ T^{lr}\ne i{d}_{A\otimes A}$ and $T^{lr}\circ R^{lr}=i{d}_{A\otimes A}$.

(ii) 

$T^{rl}\circ R^{rl}\ne i{d}_{A\otimes A}$ and $R^{rl}\circ T^{rl}=i{d}_{A\otimes A}$.

(iii) 

$\sum T^{lr}(a_{\left(1\right)}{b}_{\left(1\right)}\otimes S\left(a_{\left(2\right)}{b}_{\left(2\right)}\right)c)=\sum T^{lr}(a_{\left(1\right)}{b}_{\left(1\right)}\otimes S\left(b_{\left(2\right)}\right)S\left(a_{\left(2\right)}\right)c)$, for any $a,b,c\in H$.

Proof. 

(1)

The proof of (i)–(ii) is straightforward.

For (iii), we have, for any $a,b,c\in H$

$$\begin{array}{ccc}& & \sum T^{rl}(cS\left(a_{\left(1\right)}{b}_{\left(1\right)}\right)\otimes a_{\left(2\right)}{b}_{\left(2\right)})=c\otimes ab\mathrm{by} \mathrm{the} \mathrm{first} \mathrm{identity} \mathrm{in} \left(1\right)\left(\mathrm{ii}\right)\hfill \\ & =& (1\otimes a)(c\otimes b)=\sum (1\otimes a)T^{rl}(cS\left(b_{\left(1\right)}\right)\otimes b_{\left(2\right)})\mathrm{by} \mathrm{the} \mathrm{first} \mathrm{identity} \mathrm{in} \left(1\right)\left(\mathrm{ii}\right)\hfill \\ & =& \sum (1\otimes a)(cS\left(b_{\left(1\right)}\right)\otimes 1)\mathsf{\Delta }\left(b_{\left(2\right)}\right)\mathrm{by} \mathrm{the} \mathrm{definition} \mathrm{of}{T}^{rl}\hfill \\ & =& \sum (cS\left(b_{\left(1\right)}\right)\otimes a)\mathsf{\Delta }\left(b_{\left(2\right)}\right)\hfill \\ & =& \sum (cS\left(b_{\left(1\right)}\right)S\left(a_{\left(1\right)}\right)\otimes 1)\mathsf{\Delta }\left(a_{\left(2\right)}\right)\mathsf{\Delta }\left(b_{\left(2\right)}\right)\mathrm{by} \mathrm{the} \mathrm{definition} \mathrm{of} S\hfill \\ & =& \sum (cS\left(b_{\left(1\right)}\right)S\left(a_{\left(1\right)}\right)\otimes 1)\mathsf{\Delta }\left(a_{\left(2\right)}{b}_{\left(2\right)}\right)\mathrm{since}\mathsf{\Delta }\mathrm{is} \mathrm{a} \mathrm{bialgbera} \mathrm{map}\hfill \\ & =& \sum T^{rl}\left(cS\left(b_{\left(1\right)}\right)S\left(a_{\left(1\right)}\right)\right)\otimes a_{\left(2\right)}{b}_{\left(2\right)})\mathrm{by} \mathrm{the} \mathrm{definition} \mathrm{of}{T}^{rl}.\hfill \end{array}$$

(2)

In a similar way, we can check this.

□

As a corollary of Proposition 1, we have the main result in [3] as follows.

Corollary 1. 

A bialgebra H is a Hopf algebra if and only if the maps $T^{lr}$ and $T^{rl}$ are bijective.

Proof. 

If H is a Hopf algebra, then it follows from (1)(i) in Proposition 1 that the map $T^{lr}$ has the left inverse $R^{lr}$. It is easy to check that $R^{lr}$ is also the right inverse of $T^{lr}$, i.e., $T^{lr}\circ R^{lr}=i{d}_{H\otimes H}$. This is similar for the map $T^{rl}$. Thus, the maps $T^{lr}$ and $T^{rl}$ are bijective.

Conversely, for example for $T^{rl}$, we have $(R^{rl}\circ T^{rl})(c\otimes a)=\sum c{a}_{\left(1\right)}S\left(a_{\left(2\right)}\right)\otimes a_{\left(3\right)}=i{d}_{A\otimes A}(c\otimes a)$ for any $a,c\in H$. We can apply $\epsilon$ and obtain $\sum a_{\left(1\right)}S\left(a_{\left(2\right)}\right)=\epsilon \left(a\right)1_H$, i.e., S is also a right antipode of H.

Furthermore, notice that $T^{rl}$ being bijective implies that $\sum a_{\left(1\right)}{b}_{\left(1\right)}\otimes S\left(a_{\left(2\right)}{b}_{\left(2\right)}\right)c=\sum a_{\left(1\right)}{b}_{\left(1\right)}\otimes S\left(b_{\left(2\right)}\right)S\left(a_{\left(2\right)}\right)c$ for any $a,b,c\in H$ (actually, only by using the injection of $T^{rl}$).

We can apply $\epsilon$ and obtain $S\left(ab\right)=S\left(b\right)S\left(a\right)$, i.e., S is an algebra antihomomorphism. □

As a corollary of Proposition 1, we have

Corollary 2. 

Any left Hopf algebra is not a left multiplier Hopf algebra. Similarly, any right Hopf algebra is not a right multiplier Hopf algebra.

Let H be a left Hopf algebra. Then, we now consider the ring $End(H\otimes H)$ of all linear endomorphisms of $H\otimes H$ and two elements $T^{lr}$ and $R^{lr}$.

We introduce the following elements:

$$E_{i,j}={\left(T^{lr}\right)}^{i-1}\circ {\left(R^{lr}\right)}^{j-1}-{\left(T^{lr}\right)}^i\circ {\left(R^{lr}\right)}^j$$

(1)

for $i,j=1,2,3,\cdots ,$ where ${\left(R^{lr}\right)}^0=i{d}_{H\otimes H}={\left(T^{lr}\right)}^0$. It can be verified directly that the $E_{i,j}$ thus defined satisfy the multiplication table for matrix units: $E_{i,j}{E}_{p,q}={\delta }_{j,p}{E}_{i,q}$, where $\delta$ is the Kronecker symbol. In particular, the elements $E_i=E_{i,i}$ are orthogonal idempotent elements.

Similarly, we can consider the ring $End(H\otimes H)$ of all linear endomorphisms of $H\otimes H$ and two elements $T^{rl}$ and $R^{rl}$.

We introduce the elements

$$G_{i,j}={\left(R^{rl}\right)}^{i-1}\circ {\left(T^{rl}\right)}^{j-1}-{\left(R^{rl}\right)}^i\circ {\left(T^{rl}\right)}^j$$

(2)

for $i,j=1,2,3,\cdots ,$ where ${\left(R^{rl}\right)}^0=i{d}_{H\otimes H}={\left(T^{rl}\right)}^0$. The elements $G_i=G_{i,i}$ are orthogonal idempotent elements.

Proposition 2. 

With the above notations, $E_{i,j}\ne 0$ and $G_{i,j}\ne 0$ for any $i,j$.

Proof. 

For by $E_{i,j}{E}_{p,q}={\delta }_{j,p}{E}_{i,q}$, the vanishing of one of the $E_{i,j}$ implies the vanishing of all; in particular, it implies that $0=E_1=i{d}_{H\otimes H}-T^{lr}\circ R^{lr}$, contrary to Proposition 1(1)(i).

In the same way, we have $G_{i,j}\ne 0$ for any $i,j$. □

The existence of an infinite set of orthogonal idempotent elements in a ring R is incompatible with mild chain conditions on the ring. For the set of idempotent elements ${\left\{E_i\right\}}_{i\in \mathbb{N}}$, we set $F_k={\sum }_{i=1}^k{E}_i$, then

$$F_{1}End(H\otimes H)\subset F_{2}End(H\otimes H)\subset F_{3}End(H\otimes H)\subset \cdots$$

is an infinite properly ascending chain of right ideals. The right annihilator of an idempotent element F is the set of elements $\{X-FX\}$ for any $X\in End(H\otimes H)$. Since $End(H\otimes H)$ has the identity $i{d}_{H\otimes H}$, this right ideal is the principal right ideal $\left(i{d}_{H\otimes H}F\right)End(H\otimes H)$. It is clear that the following is an infinite properly descending chain of annihilators:

$$(i{d}_{H\otimes H}-F_1)End(H\otimes H)\supset (i{d}_{H\otimes H}-F_2)End(H\otimes H)\supset (i{d}_{H\otimes H}-F_3)End(H\otimes H)\supset \cdots .$$

By Theorem 1 in [11] and Proposition 1, we have the following result.

Proposition 3. 

Let H be a left Hopf algebra with a left antipode S.

(1) 

If $End(H\otimes H)$ satisfies either the ascending or the descending chain condition for principal right ideals generated by idempotent elements ${\left\{E_i\right\}}_{i\in \mathbb{N}}$, then $T^{lr}$ is bijective.

(2) 

If $End(H\otimes H)$ satisfies either the ascending or the descending chain condition for principal right ideals generated by idempotent elements ${\left\{G_i\right\}}_{i\in \mathbb{N}}$, then $T^{rl}$ is bijective.

Combine Corollary 1 and Proposition 3 to get:

Theorem 1. 

Let H be a left Hopf algebra with a left antipode S. If $End(H\otimes H)$ satisfies either the ascending or the descending chain condition for principal right ideals generated by idempotent elements ${\left\{E_i\right\}}_{i\in \mathbb{N}}$ and $End(H\otimes H)$ satisfies either the ascending or the descending chain condition for principal right ideals generated by idempotent elements ${\left\{G_i\right\}}_{i\in \mathbb{N}}$, then H is a Hopf algebra.

By Theorem 2 in [11] and Proposition 1, we have the following result.

Proposition 4. 

Let H be a left Hopf algebra or a right Hopf algebra. Then $End(H\otimes H)$ contains a right ideal that is a direct sum of an infinite number of $End(H\otimes H)$-isomorphic right ideals.

In what follows, we wish to determine the structures of the algebra $\mathbf{k}⟨R^{lr},T^{lr}⟩$ generated by $R^{lr}$ and $T^{lr}$ and $\mathbf{k}⟨T^{rl},R^{rl}⟩$ generated by $T^{rl}$ and $R^{rl}$.

By Theorem 4 in [11] and Proposition 1, we have the following.

Theorem 2. 

The algebras $\mathbf{k}⟨R^{lr},T^{lr}⟩$ and $\mathbf{k}⟨T^{rl},R^{rl}⟩$ are isomorphic under an isomorphism that pairs the $R^{lr}$ and the $T^{rl}$, and the $T^{lr}$ and the $R^{rl}$, respectively. The algebras $\mathbf{k}⟨R^{lr},T^{lr}⟩$ and $\mathbf{k}⟨T^{rl},R^{rl}⟩$ are primitive algebras that have minimal one-sided ideals.

It is similar for the case of right Hopf algebras. We can get the analogue of Propositions 2 and 3. Theorem 1 and Proposition 4 carry over without change.

4. Multiplier Left Hopf Algebras

In this section, let $(A,m)$ be an algebra, with a nondegenerate product m; A may or may not have an identity. We work with the following notion of a coproduct (see Definition 1.1 in [12]).

Definition 1. 

A  coproduct  (or  comultiplication) on $(A,m)$ is a homomorphism $\mathsf{\Delta }:A⟶M(A\otimes A)$ such that

(i) 

$T^{lr}(a\otimes b)\in A\otimes A$ and $T^{rl}(a\otimes b)\in A\otimes A$ for all $a,b\in A$,

(ii) 

Δ is  coassociative  in the following sense:

$$(a\otimes 1\otimes 1)\left(\right(\mathsf{\Delta }\otimes \iota \left)\right(\mathsf{\Delta }\left(b\right)(1\otimes c)\left)\right)=\left(\right(\iota \otimes \mathsf{\Delta }\left)\right((a\otimes 1)\mathsf{\Delta }\left(b\right)\left)\right)(1\otimes 1\otimes c)$$

(3)

for all $a,b,c\in A$. Moreover, coassociativity of Δ can be expressed in terms of the canonical maps as

$$(T^{rl}\otimes \iota )(\iota \otimes T^{lr})=(\iota \otimes T^{lr})(T^{rl}\otimes \iota ).$$

We remark that condition (i) makes sense because we have $A\otimes A\subseteq M\left(A\right)\otimes M\left(A\right)\subseteq M(A\otimes A)$. These conditions, in fact, imply that Δ is a nondegenerate homomorphism.

First, we want the existence of a left (right) counit.

Definition 2. 

Let Δ be a coproduct on the algebra $(A,m)$ as in Definition 1.

(i) 

A linear functional ε on $(A,m)$ is called a  left counit  if

$$(\epsilon \otimes \iota )\left(\mathsf{\Delta }\right(a\left)\right(1\otimes b\left)\right)=ab$$

for all $a,b\in A$.

(ii) 

A linear functional ε on $(A,m)$ is called a  right counit  if

$$(\iota \otimes \epsilon )\left(\right(a\otimes 1\left)\mathsf{\Delta }\right(b\left)\right)=ab$$

for all $a,b\in A$.

We now define our multiplier left (right) Hopf algebras.

Definition 3. 

Assume that $(A,m)$ is a non-degenerate algebra with a comultiplication Δ. Then

(i) 

We say that $(A,m,\mathsf{\Delta })$ is a  multiplier left Hopf algebra  if A has a right counit and the maps $T^{lr}$ is injective map and $T^{rl}$ is surjective map of $A\otimes A$ to itself. We call A  regular  if $A^{cop}$, where ${\mathsf{\Delta }}^{cop}=t\circ \mathsf{\Delta }$ for the flipping map. Then, $(A,{\mathsf{\Delta }}^{cop})$ is also a multiplier left Hopf algebra.

(ii) 

We call $(A,m,\mathsf{\Delta })$ a  multiplier right Hopf algebra  if A has a left counit and the map $T^{lr}$ is a surjective map and $T^{rl}$ is sn injective map of $A\otimes A$ to itself. We call A  regular  if $A^{cop}$ , where ${\mathsf{\Delta }}^{cop}$ is same as in (i). Then, $(A,{\mathsf{\Delta }}^{cop})$ is also a multiplier right Hopf algebra.

Clearly, we get that $(A,m,\mathsf{\Delta })$ is a multiplier left Hopf algebra if and only if $(A^{op},m,{\mathsf{\Delta }}^{cop})$ is a multiplier right Hopf algebra, where $A^{op}$ is the opposite algebra of A. For this reason, we just discuss the case of multiplier left Hopf algebras. Similarly for the right version.

Example 1. 

(1) 

A left Hopf algebra is a multiplier left Hopf algebra. Similarly, any right Hopf algebra is a multiplier right Hopf algebra.

(2) 

Any multiplier Hopf algebra is both a multiplier left Hopf algebra and a multiplier right Hopf algebra since $T^{lr}$ and $T^{rl}$ are bijective (see [3]).

Definition 4. 

If $(A,m)$ is a ∗-algebra, we say that Δ a coproduct if itself is also a ∗-homomorphism. A multiplier left (resp. right) Hopf ∗-algebra is a ∗-algebra with a comultiplication, making it into a multiplier left (resp. right) Hopf algebra.

Also, for a multiplier left (right) Hopf ∗-algebra, the regularity is automatic.

In what follows, we let $(A,m,\mathsf{\Delta })$ be a multiplier left Hopf algebra with a right counit ε. We will show that the ε is a homomorphism that has the properties of a counit in usual left Hopf algebra theory.

Following the idea of [3], however, our map $T^{rl}$ is not bijective, we have

Definition 5. 

Let $(A,m,\mathsf{\Delta })$ be a multiplier left Hopf algebra as defined in Definition 3(i). We define a map $E:A⟶R\left(A\right)$, the right multiplier algebra of A, by

$$aE\left(b\right)=\sum _i^n{a}_i{b}_{i}if{T}^{rl}(\sum a_i\otimes b_i)=a\otimes b.$$

Notice that if $a\otimes b={\sum }_i(a_i\otimes 1)\mathsf{\Delta }\left(b_i\right)={\sum }_j(p_j\otimes 1)\mathsf{\Delta }\left(q_j\right)$, then it follows from this formula and applying $\iota \otimes \epsilon$ to get $a\epsilon \left(b\right)={\sum }_i{a}_i{b}_i={\sum }_j{p}_j{q}_j$. It means that E is well-defined as right multipliers.

In the following proposition, we show that this right multiplier is actually scalar multiples of the identity.

Lemma 1. 

With the notation E as above, we have

$$(E\otimes \iota )\left(\mathsf{\Delta }\right(a\left)\right(1\otimes b\left)\right)=1\otimes ab$$

for all $a,b\in A$.

Proof. 

Let $a,b,c\in A$ and let f be any linear functional on A. Assume that $c\otimes a={\sum }_i^n(c_i\otimes 1)\mathsf{\Delta }\left(a_i\right)$. Then, we do a calculation as follows:

$$\begin{array}{ccc}\hfill c\otimes (\iota \otimes f)\left(\mathsf{\Delta }\right(a\left)\right(1\otimes b\left)\right)& =& \sum (\iota \otimes \iota f)\left((\iota \otimes \mathsf{\Delta })\left((c_i\otimes 1)\mathsf{\Delta }\left(a_i\right)\right)\right)(1\otimes 1\otimes b)\hfill \\ & \stackrel{\left(3.1\right)}{=}& (\iota \otimes \iota f)\sum (c_i\otimes 1\otimes 1)(\mathsf{\Delta }\otimes \iota )\left(\mathsf{\Delta }\left(a_i\right)(1\otimes b)\right)\hfill \\ & =& \sum (c_i\otimes 1)\mathsf{\Delta }\left((\iota \otimes f)\left(\mathsf{\Delta }\left(a_i\right)(1\otimes b)\right)\right)\hfill \\ & =& T^{rl}(c_i\otimes \sum _i(\iota \otimes f)\left(\mathsf{\Delta }\left(a_i\right)(1\otimes b)\right)).\hfill \end{array}$$

By the definition of E, we obtain

$$cE\left((\iota \otimes f)\left(\mathsf{\Delta }\left(a\right)(1\otimes b)\right)\right)=c_i\sum _i(\iota \otimes f)\left(\mathsf{\Delta }\left(a_i\right)(1\otimes b)\right).$$

Therefore, we have

$$\begin{array}{ccc}\hfill (1\otimes f)\left(\right(c\otimes 1\left)\right(E\otimes \iota \left)\right(\mathsf{\Delta }\left(a\right)(1\otimes b)\left)\right)& =& (\iota \otimes f)\left(\sum _i\left((c_i\otimes 1)\mathsf{\Delta }\left(a_i\right)\right)(1\otimes b)\right)\hfill \\ & =& (\iota \otimes f)\left(\right(c\otimes a\left)\right(1\otimes b\left)\right)\hfill \\ & =& (\iota \otimes f)\left(\right(c\otimes 1\left)\right(1\otimes ab\left)\right).\hfill \end{array}$$

Because this hold for all f, we obtain

$$(c\otimes 1)\left(\right(E\otimes \iota \left)\right(\mathsf{\Delta }\left(a\right)(1\otimes b)\left)\right)=(c\otimes 1)(1\otimes ab).$$

This gives the required formula in the multiplier algebra. □

Proposition 5. 

With the above notations, for all $a,b\in A$, we have

(i) 

$E\left(A\right)\subset \mathbf{k}1$ and $E\left(a\right)=\epsilon \left(a\right)1$.

(ii) 

$(\epsilon \otimes \iota )\left(\mathsf{\Delta }\right(a\left)\right(1\otimes b\left)\right)=ab$.

(iii) 

If ε is also a right counit, then $\epsilon \left(ab\right)=\epsilon \left(a\right)\epsilon \left(b\right)$.

Proof. 

(i)

From the definition of E, we have $aE\left(b\right)=a\epsilon \left(b\right)$ for all $a,b\in A$. This gives the result.

(ii)

By Lemma 1, for all $a,b\in A$ we have $(E\otimes \iota )\left(\mathsf{\Delta }\right(a\left)\right(1\otimes b\left)\right)=1\otimes ab$. It implies that

$$(\epsilon \otimes \iota )\left(\mathsf{\Delta }\right(a\left)\right(1\otimes b\left)\right)=ab.$$

(iii)

Since $(\iota \otimes \epsilon )\left(\right(a\otimes 1\left)\mathsf{\Delta }\right(bc\left)\right)=abc$, for all $a,b,c\in A$. Then

$$(\iota \otimes \epsilon )\left(\right(a\otimes 1\left)\mathsf{\Delta }\right(b\left)\mathsf{\Delta }\right(c\left)\right)=\left(ab\right)c=(\iota \otimes \epsilon )\left(\right(a\otimes 1\left)\mathsf{\Delta }\right(b\left)\right)c.$$

By the surjectivity of $T^{rl}$, we get

$$\begin{array}{ccc}& & (\iota \otimes \epsilon )\left(\right(a\otimes b\left)\mathsf{\Delta }\right(c\left)\right)=(\iota \otimes \epsilon )(a\otimes b)c\hfill \\ & =& a\epsilon \left(b\right)c=\epsilon \left(b\right)ac\hfill \\ & =& \epsilon \left(b\right)(\iota \otimes \epsilon )\left(\right(a\otimes 1\left)\mathsf{\Delta }\right(c\left)\right),\hfill \end{array}$$

and $(\iota \otimes \epsilon )(a\otimes bc)=\epsilon \left(b\right)(\iota \otimes \epsilon )(a\otimes c)$.

This means $a\epsilon \left(bc\right)=a\epsilon \left(b\right)\epsilon \left(c\right)$. □

Therefore, altogether we obtain the following result.

Theorem 3. 

Let A be a multiplier left Hopf algebra with a right counit ε. Then, ε is an algebra homomorphism such that

$$(\epsilon \otimes \iota )T^{lr}(a\otimes b))=ab,and(\iota \otimes \epsilon )T^{rl}(a\otimes b)=ab$$

for all $a,b\in A$.

It is clear that if A has an identity, then ε is a counit in the usual sense.

We now show that a left antipode exists with the expected properties of the left antipode in the usual left Hopf algebra theory.

Definition 6. 

Define a map $S:A⟶R\left(A\right)$ by

$$aS\left(b\right)=(\iota \otimes \epsilon ){\left(T^{rl}\right)}^{-1}(a\otimes b)$$

for all $a,b\in A$.

We can rewrite these formulas as

$$aS\left(b\right)=\sum _i{x}_i\epsilon \left(y_i\right)ifa\otimes b=\sum _i(x_i\otimes 1)\mathsf{\Delta }\left(y_i\right)=T^{rl}(\sum x_i\otimes y_i).$$

As before, from this formula, it is easy to see that, indeed, $S\left(b\right)$ is well-defined linear maps from A to the algebra $R\left(A\right)$ of right multipliers for all $b\in A$.

Proposition 6. 

For all $a,b\in A$, we have that $(a\otimes 1)\left(\right(S\otimes \iota \left)\mathsf{\Delta }\right(b\left)\right)\in A\otimes A$ and

$$m\left(\right(a\otimes 1\left)\right((S\otimes \iota )\mathsf{\Delta }\left(b\right)\left)\right)=\epsilon \left(b\right)a.$$

Proof. 

One begins as in the proof of Lemma 1. For $a,b,c\in A$ and set $c\otimes a={\sum }_i^n(c_i\otimes 1)\mathsf{\Delta }\left(a_i\right)$, we have

$$\begin{array}{ccc}\hfill (c\otimes 1)(S\otimes \iota )\mathsf{\Delta }\left(a\right)(1\otimes b))& =& \sum (c_i\otimes 1\otimes 1)(S\otimes \iota )(\mathsf{\Delta }\otimes \iota )\left(\mathsf{\Delta }\left(a_i\right)(1\otimes b)\right)\hfill \\ & =& \sum (c_i\otimes 1)(\epsilon \otimes \iota )\left(\mathsf{\Delta }\left(a_i\right)(1\otimes b)\right)\mathrm{by} \mathrm{the} \mathrm{definition} \mathrm{of} S\hfill \\ & =& \sum _i{c}_i\otimes a_{i}b.\hfill \end{array}$$

It follows that $(c\otimes 1)\left(\right(S\otimes \iota \left)\mathsf{\Delta }\right(a\left)\right)\in A\otimes A$ and that

$$m(c\otimes 1)\left(\right(S\otimes \iota \left)\mathsf{\Delta }\right(a\left)\right)=\epsilon \left(a\right)c.$$

□

Using the Sweedler notation, one gets

$$\sum _{\left(b\right)}aS\left(b_{\left(1\right)}\right)b_{\left(2\right)}=\epsilon \left(b\right)a$$

(4)

From the result above, we have that

$$\sum _{\left(b\right)}aS\left(b_{\left(1\right)}\right)\otimes b_{\left(2\right)}$$

is a well-defined element in $A\otimes A$. This is what we need for the formula in Equation (4).

Then, we have the following main result.

Theorem 4. 

If $(A,m,\mathsf{\Delta },S)$ is a multiplier left Hopf algebra, then there exists linear map $S:A⟶R\left(A\right)$ such that

$$m\left(\right(c\otimes 1\left)\right(S\otimes \iota \left)\right(\mathsf{\Delta }\left(a\right)(1\otimes b)\left)\right)=c\epsilon \left(a\right)b$$

for all $a,b,c\in A$.

It is not so hard to show that the above formulas determine S, just as in the case of the counit. If A has an identity, then we have the usual formula

$$m(S\otimes \iota )\mathsf{\Delta }\left(a\right)=\epsilon \left(a\right)1.$$

If we combine this with the results on ε, we obtain the following theorem.

Theorem 5. 

If A is a multiplier left Hopf algebra with an identity, then A is a left Hopf algebra.

Example 2. 

Let A be a left Hopf algebra with a left antipode $S_A$ and B a multiplier Hopf algebra with an antipode $S_B$. Then, the tensor product algebra $A\otimes B$ can be made into a multiplier left Hopf algebra with the following structures:

$$\left\{\begin{array}{cc}The product:\hfill & (a\otimes b)(x\otimes y)=ax\otimes by,for alla,x\in A,b,y\in B;\hfill \\ The coproduct:\hfill & \mathsf{\Delta }(a\otimes b)=\sum (a_{\left(1\right)}\otimes b_{\left(1\right)})\otimes (a_{\left(2\right)}\otimes b_{\left(2\right)}),for alla\in A,b\in B;\hfill \\ The counit:\hfill & \epsilon (a\otimes b)=\epsilon \left(a\right)\epsilon \left(b\right),for alla\in A,b\in B;\hfill \\ The antipode:\hfill & S(a\otimes b)=S_A\left(a\right)\otimes S_B\left(b\right),for alla\in A,b\in B.\hfill \end{array}\right.$$

We notice that

$$\begin{array}{ccc}\hfill T^{lr}\left((a\otimes b)(x\otimes y)\right)& =& \mathsf{\Delta }(a\otimes b)(1\otimes 1\otimes x\otimes y)\hfill \\ & =& \sum (a_{\left(1\right)}\otimes b_{\left(1\right)})\otimes (a_{\left(2\right)}x\otimes b_{\left(2\right)}y)\in (A\otimes B)\otimes (A\otimes B)\hfill \end{array}$$

and, similarly,

$$T^{rl}\left((a\otimes b)(x\otimes y)\right)\in (A\otimes B)\otimes (A\otimes B).$$

It is easy to check that

$$(T^{rl}\otimes \iota \otimes \iota )(\iota \otimes \iota \otimes T^{lr})=(\iota \otimes \iota \otimes T^{lr})(T^{rl}\otimes \iota \otimes \iota ).$$

Since $S_A$ is a left antipode with $\sum S_A\left(a_{\left(1\right)}\right)a_{\left(2\right)}=\epsilon \left(a\right)1_A$ and by Proposition 6, we have

$$\sum S_A\left(a_{\left(1\right)}{a}_{\left(2\right)}\right)\otimes y{S}_B(b_{\left(1\right)}{b}_{\left(2\right)}=\epsilon \left(a\right)1\otimes \epsilon \left(b\right)y,foralla\in A,b,y\in B.$$

5. Multiplier Hopf Algebras and Multiplier Left Hopf Algebras

Let A be an algebra with a nondegenerate product and with or without an identity. In this section we mainly generalize the results obtained in the section to the case of multiplier algebras. Therefore, we give a sufficient condition for a multiplier left Hopf algebra to be a multiplier Hopf algebra.

Let $\mathsf{\Delta }$ be a coproduct on A. From Definition 1, we have that $T^{lr}(a\otimes b)\in A\otimes A$ and $T^{rl}(a\otimes b)\in A\otimes A$ for all $a,b\in A$. We can define $E_{i,j}$ and $G_{i,j}$ as in Equations (1) and (2), respectively.

Similar to Proposition 3, we have

Proposition 7. 

Let A be a multiplier left Hopf algebra.

(1) 

If $End(A\otimes A)$ satisfies either the ascending or the descending chain condition for principal right ideals generated by idempotent elements ${\left\{E_i\right\}}_{i\in \mathbb{N}}$, then $T^{lr}$ is bijective.

(2) 

If $End(A\otimes A)$ satisfies either the ascending or the descending chain condition for principal right ideals generated by idempotent elements ${\left\{G_i\right\}}_{i\in \mathbb{N}}$, then $T^{rl}$ is bijective.

Similar to Theorem 1, we have

Theorem 6. 

Let A be a multiplier left Hopf algebra. If $End(A\otimes A)$ satisfies either the ascending or the descending chain conditions for principal right ideals both generated by idempotent elements ${\left\{E_i\right\}}_{i\in \mathbb{N}}$ and generated by idempotent elements ${\left\{G_i\right\}}_{i\in \mathbb{N}}$, then A is a multiplier Hopf algebra.

Finally, the following is easy to see.

Proposition 8. 

(i) 

Any multiplier left Hopf algebra is not a left multiplier Hopf algebra. Conversely, any left multiplier Hopf algebra is a multiplier left Hopf algebra.

(ii) 

Similarly, any multiplier right Hopf algebra is not a right multiplier Hopf algebra. Conversely, any right multiplier Hopf algebra is a multiplier right Hopf algebra.

Proof. 

(i)

If A is a multiplier left Hopf algebra, then we cannot get that $T^{ll}$ is bijective. Indeed, we cannot get that $T^{lr}$ is bijective. Conversely, it follows from Theorem 3.8 in [9] that any left multiplier Hopf algebra is a regular multiplier Hopf algebra. Then, it follows Example 1(2).

(ii)

For this, we have a similar statement to the one in (i).

□

Proposition 9. 

Let A be a multiplier left Hopf algebra that is left or right Noetherian as an algebra. Then, A is a multiplier Hopf algebra.

Proof. 

First, assume that A is left Noetherian. For $a\in A$, we write $\mathsf{\Delta }\left(a\right)={\sum }_i{x}_i\otimes y_i$. Set $V={\sum }_{m,n=0}^{\infty }{\left(R^{rl}\right)}^n{\left(T^{rl}\right)}^m$, with $R^{rl}$ and $T^{rl}$ as above; V is clearly $R^{rl}$-stable, and, since $T^{rl}{R}^{rl}=i{d}_{A\otimes A}$, V is also $T^{rl}$-stable. We have that

$${\left(R^{rl}\right)}^n{\left(T^{rl}\right)}^m(a\otimes b)=a{b}_{\left(1\right)}{b}_{\left(2\right)}\cdots b_{\left(m\right)}S\left(b_{(m+1)}\right)S\left(b_{(m+2)}\right)\cdots S\left(b_{(m+n)}\right)\otimes b_{(m+n+1)}.$$

Since ${\mathsf{\Delta }}_{n+m}\left(a\right)={\sum }_i{\mathsf{\Delta }}_{n+m-1}\otimes y_i$, we have that $V\subset {\sum }_{i}A\otimes y_i$. (Note that for $n=m=0,a\otimes b=a\otimes {\sum }_i\epsilon \left(x_i\right)y_i={\sum }_i\epsilon \left(x_i\right)a\otimes y_i$). Since A is left Noetherian, V is a Noetherian left A-module. As before, $T^{rl}\left(V\right)\subset V,R^{rl}\left(V\right)\subset V,T^{rl}{R}^{rl}=i{d}_{A\otimes A}$ imply $T^{rl}\left(V\right)=V$, so the restriction of $T^{rl}$ to V is injective. This implies that $T^{rl}$ is injective. For if ${\sum }_i{a}_i\otimes b_i\in A\otimes A$, each $A\otimes b_i$ is contained in a finitely generated left A-submodule $V_i$ of $A\otimes A$ for which $T^{rl}\left(V_i\right)=V_i$. Therefore, ${\sum }_i{a}_i\otimes b_i\in V={\sum }_i{V}_i$, V is a finitely generated left A-module with $T^{rl}\left(V\right)=V$. Since V is Noetherian, $T^{rl}$ is injective on V as above. Thus, $R^{rl}={\left(T^{rl}\right)}^{-1}$, and we continue as before.

If A is right Noetherian, we consider $A\otimes A$ as a right A-module and consider $T^{lr}$ and $R^{lr}$. Then, $R^{lr}{T}^{lr}=i{d}_{A\otimes A}$. We set $V={\sum }_{m,n=0}^{\infty }{\left(T^{lr}\right)}^n{\left(R^{lr}\right)}^m$ and proceed as above. □

6. Examples

We give some examples in this section to illustrate various notions and results obtained in the previous sections. In particular, we provide insight into previously given theorems.

Recall from [13] or [14] that the bialgebra $L_q$, $q\ne 0$ in $\mathbf{k}$ is a free noncommutative algebra on the $\left\{X_{ij}\right\},i,j=1,2$, with the following three relations:

$$\begin{array}{ccc}& & X_{21}{X}_{11}=q{X}_{11}{X}_{21},\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & X_{22}{X}_{12}=q{X}_{12}{X}_{22},\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& & X_{22}{X}_{11}=X_{11}{X}_{22}-q^{-1}{X}_{21}{X}_{12}+q{X}_{12}{X}_{21},\hfill \end{array}$$

(7)

with coalgebra structure given by $\mathsf{\Delta }{X}_{ij}={\sum }_{k=1}^2{X}_{ik}\otimes X_{kj}$ and $\epsilon \left(X_{ij}\right)={\delta }_{ij}$. The quantum determinant $d=X_{11}{X}_{22}-q^{-1}{X}_{21}{X}_{12}=X_{22}{X}_{11}-q{X}_{12}{X}_{21}$ is group-like but no longer central. We can invert d by introducing a non-central variable y and imposing the relations $dy=1$ and $yd=1$. Let X be the generic matrix $\left(X_{ij}\right)$. Then, the left antipode S is given by

$$S\left(X\right)=X^{-1}=y\left(\begin{array}{cc}{X}_{22}& -q{X}_{12}\\ -q^{-1}{X}_{21}& X_{11}\end{array}\right)$$

and $S\left(y\right)=d$.

Recall from [4] that a multiplier Hopf algebra $A(m,n)$ is a nonunital associative algebra generated by a linear basis ${\left\{Y_{m,n}\right\}}_{m\in \mathbb{Z},n=0,1}$ with the product induced by the following commutation rules:

$$\begin{array}{ccc}& & Y_{p,0}{Y}_{q,0}={\delta }_{p,q}{Y}_{p,0},Y_{p,0}{Y}_{q,1}={\delta }_{p-q,1}{Y}_{q,1},\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& & Y_{q,1}{Y}_{p,0}={\delta }_{p,q}{Y}_{p,1},Y_{p,1}{Y}_{q,1}=0\hfill \end{array}$$

(9)

where $p,q\in \mathbb{Z}$ and ${\delta }_{p,q}$ denotes the Kronecker-delta symbol.

The unit in $M\left(A\right(m,n\left)\right)$ is given as $1={\sum }_{p\in \mathbb{Z}}{Y}_{p,0}$. The coproduct, counit, and antipode on $A(m,n)$ are given as follows:

$$\begin{array}{ccc}& & \mathsf{\Delta }\left(Y_{p,0}\right)=\sum _{q\in \mathbb{Z}}{Y}_{q,0}\otimes Y_{p-q,0},\epsilon \left(Y_{p,0}\right)={\delta }_{p,0};\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & \mathsf{\Delta }\left(Y_{p,1}\right)=\sum _{r\in \mathbb{Z}}{Y}_{r,0}\otimes Y_{p-r,1}+\sum _{r\in \mathbb{Z}}{(-1)}^{p-r}{Y}_{r,1}\otimes Y_{p-r,0},\epsilon \left(Y_{p,1}\right)=0;\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & S\left(Y_{p,0}\right)=Y_{-p,0},S\left(Y_{p,1}\right)={(-1)}^p{Y}_{-p-1,1}.\hfill \end{array}$$

(12)

By Example 2, we have

Example 3. 

With the above notations. $H_q(m,n):=L_q\otimes A(m,n)$ is a multiplier left Hopf algebra with the following structures:

• The product: for all $X_{ij},X_{i^{\prime }{j}^{\prime }}\in L_q,{\left\{Y_{m,n}\right\}}_{m\in \mathbb{Z},n=0,1}\in A(m,n)$,

  $$\begin{array}{ccc}& & (X_{ij}\otimes Y_{p,0})(X_{i^{\prime }{j}^{\prime }}\otimes Y_{q,0})=X_{ij}{X}_{i^{\prime }{j}^{\prime }}\otimes {\delta }_{p,q}{Y}_{p,0},\hfill \\ & & (X_{ij}\otimes Y_{p,0})(X_{i^{\prime }{j}^{\prime }}\otimes Y_{q,1})=X_{ij}{X}_{i^{\prime }{j}^{\prime }}\otimes {\delta }_{p-q,1}{Y}_{q,1},\hfill \\ & & (X_{ij}\otimes Y_{q,1})(X_{i^{\prime }{j}^{\prime }}\otimes Y_{p,0})=X_{ij}{X}_{i^{\prime }{j}^{\prime }}\otimes {\delta }_{p,q}{Y}_{p,1},\hfill \\ & & (X_{ij}\otimes Y_{p,1})(X_{i^{\prime }{j}^{\prime }}\otimes Y_{q,1})=0.\hfill \end{array}$$
• The coproduct: for all $X_{ij}\in L_q,{\left\{Y_{m,n}\right\}}_{m\in \mathbb{Z},n=0,1}\in A(m,n)$,

  $$\begin{array}{ccc}\hfill \mathsf{\Delta }(X_{ij}\otimes Y_{p,0})& =& \sum _{q\in \mathbb{Z}}\sum _{k=1}^2(X_{ik}\otimes Y_{q,0})\otimes (X_{kj}\otimes Y_{p-q,0}),\hfill \\ \hfill \mathsf{\Delta }(X_{ij}\otimes Y_{p,1})& =& \sum _{r\in \mathbb{Z}}\sum _{k=1}^2((X_{ik}\otimes Y_{r,0})\otimes (X_{kj}\otimes Y_{p-r,1})\hfill \\ & +& (X_{ik}\otimes {(-1)}^{p-r}{Y}_{r,1})\otimes (X_{kj}\otimes Y_{p-r,0})).\hfill \end{array}$$
• The counit: for all $X_{ij}\in L_q,{\left\{Y_{m,n}\right\}}_{m\in \mathbb{Z},n=0,1}\in A(m,n)$,

  $$\begin{array}{ccc}& & \epsilon (X_{ij}\otimes Y_{p,0})={\delta }_{i,j}{\delta }_{p,0},\hfill \\ & & \epsilon (X_{ij}\otimes Y_{p,1})=0.\hfill \end{array}$$
• The antipode: for all $X_{ij}\in L_q,{\left\{Y_{m,n}\right\}}_{m\in \mathbb{Z},n=0,1}\in A(m,n)$,

  $$\begin{array}{cc}& S(X_{ij}\otimes Y_{p,0})=S\left(X_{ij}\right)\otimes Y_{-p,0},\\ & S(X_{ij}\otimes Y_{p,1})=S\left(X_{ij}\right)\otimes {(-1)}^p{Y}_{-p-1,1}.\end{array}$$

We can compute as follows:

$$\begin{array}{ccc}& & T^{lr}\left((X_{ij}\otimes Y_{p,0})(X_{i^{\prime }{j}^{\prime }}\otimes Y_{q,0})\right)\hfill \\ & =& \mathsf{\Delta }(X_{ij}\otimes Y_{p,0})(1\otimes 1\otimes X_{i^{\prime }{j}^{\prime }}\otimes Y_{q,0})\hfill \\ & =& \sum _{r\in \mathbb{Z}}\sum _{k=1}^2((X_{ik}\otimes Y_{r,0})\otimes (X_{kj}\otimes Y_{p-r,0}))(1\otimes 1\otimes X_{i^{\prime }{j}^{\prime }}\otimes Y_{q,0})\hfill \\ & & by the definition of\mathsf{\Delta }\hfill \\ & =& \sum _{r\in \mathbb{Z}}\sum _{k=1}^2\left((X_{ik}\otimes Y_{r,0})\right)\otimes (X_{kj}{X}_{i^{\prime }{j}^{\prime }}\otimes Y_{p-r,0}{Y}_{q,0})\hfill \\ & =& \sum _{r\in \mathbb{Z}}\sum _{k=1}^2\left((X_{ik}\otimes Y_{r,0})\right)\otimes (X_{kj}{X}_{i^{\prime }{j}^{\prime }}\otimes {\delta }_{p-r,q}{Y}_{p-r,0})\hfill \\ & & by the definition of the product\hfill \\ & =& \sum _{k=1}^2(X_{ik}\otimes Y_{p-q,0})\otimes (X_{kj}{X}_{i^{\prime }{j}^{\prime }}\otimes Y_{q,0})\hfill \\ & \in & (L_q\otimes A(m,n))\otimes (L_q\otimes A(m,n)).\hfill \end{array}$$

In a same way, we can get

$$\begin{array}{ccc}\hfill T^{lr}\left((X_{ij}\otimes Y_{p,0})(X_{i^{\prime }{j}^{\prime }}\otimes Y_{q,1})\right)& =& \mathsf{\Delta }(X_{ij}\otimes Y_{p,0})(1\otimes 1\otimes X_{i^{\prime }{j}^{\prime }}\otimes Y_{q,1})\hfill \\ & =& \sum _{k=1}^2(X_{ik}\otimes Y_{p-q-1,0})\otimes (X_{kj}{X}_{i^{\prime }{j}^{\prime }}\otimes Y_{q,1})\hfill \\ & \in & (L_q\otimes A(m,n))\otimes (L_q\otimes A(m,n)),\hfill \end{array}$$

and

$$\begin{array}{ccc}& & T^{lr}\left((X_{ij}\otimes Y_{q,1})(X_{i^{\prime }{j}^{\prime }}\otimes Y_{p,0})\right)\hfill \\ & =& \sum _{k=1}^2(X_{ik}\otimes Y_{q-p,0})\otimes (X_{kj}{X}_{i^{\prime }{j}^{\prime }}\otimes Y_{p,1})+(X_{ik}\otimes {(-1)}^p{Y}_{q-p,1})\otimes (X_{kj}{X}_{i^{\prime }{j}^{\prime }}\otimes Y_{p,0})\hfill \\ & \in & (L_q\otimes A(m,n))\otimes (L_q\otimes A(m,n))\hfill \end{array}$$

and $T^{lr}(X_{ij}\otimes Y_{p,1})(X_{i^{\prime }{j}^{\prime }}\otimes Y_{q,1})=0\in (L_q\otimes A(m,n))\otimes (L_q\otimes A(m,n))$.

Similarly, we can compute the following items:

$$\begin{array}{ccc}& & T^{rl}\left((X_{ij}\otimes Y_{p,0})(X_{i^{\prime }{j}^{\prime }}\otimes Y_{q,0})\right),T^{rl}\left((X_{ij}\otimes Y_{p,0})(X_{i^{\prime }{j}^{\prime }}\otimes Y_{q,1})\right),\hfill \\ & & T^{rl}\left((X_{ij}\otimes Y_{q,1})(X_{i^{\prime }{j}^{\prime }}\otimes Y_{p,0})\right),T^{rl}\left((X_{ij}\otimes Y_{p,1})(X_{i^{\prime }{j}^{\prime }}\otimes Y_{q,1})\right).\hfill \end{array}$$

It is easy to check that

$$(T^{rl}\otimes \iota \otimes \iota )(\iota \otimes \iota \otimes T^{lr})=(\iota \otimes \iota \otimes T^{lr})(T^{rl}\otimes \iota \otimes \iota ).$$

Since $S_{L_q}$ is a left antipode and from Proposition 6, we have

$$\sum S_{L_q}\left({X_{ij}}_{\left(1\right)}{X_{ij}}_{\left(2\right)}\right)\otimes yS\left({Y_{m,n}}_{\left(1\right)}\right){Y_{m,n}}_{\left(2\right)}=\epsilon \left(X_{ij}\right)1\otimes \epsilon \left(Y_{m,n}\right)y,$$

for all $X_{ij}\in L_q,y,{\left\{Y_{m,n}\right\}}_{m\in \mathbb{Z},n=0,1}\in A(m,n)$, where S is the antipode of $A(m,n)$.