Twisted Algebras of Multiplier Hopf ($^*$-)algebra

TL;DR

This paper explores twisted algebra structures within multiplier Hopf ($^*$-)algebras, unifying various smash product constructions in the context of advanced algebraic frameworks.

Contribution

It introduces a generalized framework for twisted algebras that encompasses multiple existing smash product types in multiplier Hopf ($^*$-)algebras.

Findings

Unified approach to various smash products

Extension of algebraic structures in multiplier Hopf algebras

New insights into twisted algebra constructions

Abstract

In this paper we study twisted algebras of multiplier Hopf ($^*$-)algebras which generalize all kinds of smash products such as generalized smash products, twisted smash products, diagonal crossed products, L-R-smash products, two-sided crossed products and two-sided smash products for the ordinary Hopf algebras appeared in [P-O].

Equations

$$(1\otimes 1\otimes x)((\iota \otimes {\rm Y})(\Gamma (a)(y\otimes 1)))=((\Gamma \otimes \iota )((1\otimes x){\rm Y}(a)))(y\otimes 1)$$

$$⟨x,\hat{x}⟩\in \mathbb{C},x\in Q,\text{mbox}and\hat{x}\in \hat{Q}.$$

$$\hat{x}⊳v=(\iota \otimes \hat{x}){\rm Y}(v)$$

$$v⊲\hat{x}=(\hat{x}\otimes \iota )\Gamma (v).$$

$$(2.1)$$

$$(2.2)$$

$$(2.3)$$

$$(2.4)$$

$$(2.5)$$

$$m_{A{\#}_{RT}B}=(m_A\otimes m_B)\circ T_{14}\circ R_{23}\text{mbox}or(a\#b)(c\#d)=a_T{c}_R\#b_R{d}_T$$

$$[(a\#b)(c\#d)](x\#y)$$

$$(a\#b)[(c\#d)(x\#y)]$$

$$m_{A{\#}_{R}B}=(m_A\otimes m_B)\circ R_{23}\text{mbox}or(a\#b)(c\#d)=a{c}_R\otimes b_{R}d$$

$$(R\ast T)(a\otimes x\otimes b\otimes y)=(a{b}_R{)}_T\otimes (x{y}_T{)}_R;$$

$$(T\ast R)(b\otimes y\otimes a\otimes x)=(b_{T}a{)}_R\otimes (y_{R}x{)}_T$$

$$\sum a_{iT}{a}_r\#b_{ir}{b}_T=0,$$

$$\sum (a_{iT{R}^{-1}}a{)}_R\#b_{i{R}^{-1}R}{b}_T=0,$$

$$\sum (a_{i{t}^{-1}{R}^{-1}T}a{)}_R\#(b_{i{t}^{-1}{R}^{-1}R}b{)}_T=0.$$

$$\sum a_{i{t}^{-1}{R}^{-1}}\#b_{i{t}^{-1}{R}^{-1}}=0.$$

$$(2.6)$$

$$(2.7)$$

$$(2.8)$$

$$(2.9)$$

$$(2.10)$$

$$(a\#b)(c\#d)=a_T{c}_R\#b_R{d}_T,$$

$${\Delta }_{RT}=(i{d}_A\otimes \tau \otimes i{d}_B)({\Delta }_A\otimes {\Delta }_B),$$

$${\epsilon }_{RT}={\epsilon }_A\otimes {\epsilon }_B,$$

$$(2.11)$$

$$(2.12)$$

$$S_{RT}=TR(S_B\otimes S_A)\tau .$$

$${\Delta }_{RT}[(a\#b)(c\#d)]$$

$$={\Delta }_{RT}(a_T{c}_R\#b_R{d}_T)$$

$$=(i{d}_A\otimes \tau \otimes i{d}_B)({\Delta }_A\otimes {\Delta }_B)(a_T{c}_R\#b_R{d}_T)$$

$$=\sum (a_T{c}_R{)}_1\#(b_R{d}_T{)}_1\otimes (a_T{c}_R{)}_2\#(b_R{d}_T{)}_2$$

$$=\sum a_{T1}{c}_{R1}\#b_{R1}{d}_{T1}\otimes a_{T2}{c}_{R2}\#b_{R2}{d}_{T2}$$

$$\stackrel{(2.9)(2.10)}{=}\sum a_{1T}{c}_{1R}\#b_{1R}{d}_{1T}\otimes a_{2T}{c}_{2R}\#b_{2R}{d}_{2T}$$

$$={\Delta }_{RT}(a_T{c}_R\#b_R{d}_T){\Delta }_{RT}(a_T{c}_R\#b_R{d}_T).$$

$$m_{A{\#}_{RT}B}(S_{RT}\otimes i{d}_{A\otimes B})({\Delta }_{RT}(a\#b))$$

$$=\sum S(a_1{)}_{RTt}{a}_{2r}\#S(b_1{)}_{RTr}{b}_{2t}$$

$$\stackrel{(2.5)}{=}\sum S(a_1{)}_{RTt}{a}_{2r}\#S(b_1{)}_{RrT}{b}_{2t}$$

Full text

Twisted Algebras of Multiplier Hopf (∗-)algebras

**Shuanhong Wang

**Department of Mathematics, Southeast University

Nanjing, Jiangsu 210096, P. R. of China

E-mail: shuanhwang@seu.edu.cn

ABSTRACT

In this paper we study twisted algebras of multiplier Hopf (∗-)algebras which generalize all kinds of smash products such as generalized smash products, twisted smash products, diagonal crossed products, L-R-smash products, two-sided crossed products and two-sided smash products for the ordinary Hopf algebras appeared in [P-O].

Mathematics Subject Classifications (2000): 16W30.

Key words: Twisted tensor product; Twisted algebra; Multiplier Hopf (∗-)algebra.

1. Introduction

In [P-O], Panaite and Van Oystaeyen introduced a more general version of the so-called L-R-smash product for (quasi-)Hopf algebras and studied its relations with other kinds of crossed products (two-sided smash and crossed product and diagonal crossed product), In order to find the method of constructing multiplier Hopf (∗-)algebras, in this paper we will try to generalize these results to the situation for multiplier Hopf algebras introduced in [VD1-VD3].

This paper is organized as follows. In Section 2 of this paper we will study the notions of a generalized twisted tensor product, a generalized twisted smash product and a generalized L-R-smash product. We also study some isomorphisms between them. In Section 3 we focus on two-times twisted tensor products for multiplier Hopf algebras. Finally, in Section 4 we will introduce the notion of a Long module algebra and study the the twisted product of the given multiplier Hopf algebras.

1. Preliminaries

Let $A$ be an associative algebra with a nondegenerate product. If $A$ has a unit element, this requirement is automatically satisfied. In the general case, we only suppose the existence of local units in the following sense. Let $\{e_{1},\cdots,e_{n}\}$ be a finite set of elements in $A$. Then there exist elements $e,f\in A$ so that $ee_{i}=e_{i}=e_{i}f$ for $i=1,\cdots,n$.

A multiplier $m=(m_{1},m_{2})$ of the algebra $A$ is a pair of linear mappings in $End_{k}(A)$ such that $m_{2}(a)b=am_{1}(b)$ for all $a,b\in A$. The set of multipliers of $A$ is denoted by $M(A)$. It is a unital algebra which contains $A$ as essential ideal through the embedding $a\hookrightarrow(a\cdot,\,\cdot a)$. Therefore $m\cdot a=(m_{1}(a)\cdot,\,\cdot m_{1}(a))\equiv m_{1}(a)$ and $a\cdot m=(m_{2}(a)\cdot,\,\cdot m_{2}(a))\equiv m_{2}(a)$ for all $m\in M(A)$ and $a\in A$. Hence we will frequently use the identification $a\cdot m=m_{2}(a)$ and $m\cdot a=m_{1}(a)$. If $A$ is unital then $A=M(A)$. If $A$ is a $*$-algebra then $M(A)$ is a $*$-algebra through $m^{*}=(m_{2}^{*},\,m_{1}^{*})$ where $\psi^{*}(a):=\psi(a^{*})^{*}$ for any $a\in A$, $\psi\in End_{k}(A)$. Since the multiplication of $A$ is supposed to be nondegenerate a multiplier $m=(m_{1},m_{2})$ of $A$ is uniquely determined by its first or second component. For a tensor product of two algebras $A$ and $B$ one obtains the canonical algebra embeddings $A\otimes B\hookrightarrow M(A)\otimes M(B)\hookrightarrow M(A\otimes B)$.

Let $A$ be an algebra with a non-degenerate product. An algebra homomorphism $\Delta:A\longrightarrow M(A\otimes A)$ is called a comultiplication on $A$ if

(i) $T_{1}(x\otimes y)=\Delta(x)(1\otimes y)\in A\otimes A$ and $T_{2}(x\otimes y)=(x\otimes 1)\Delta(y)\in A\otimes A$ for all $x,y\in A$ (some times we call this condition a ”coving theory”);

(ii) $(T_{2}\otimes id)(id\otimes T_{1})=(id\otimes T_{1})(T_{2}\otimes id)$ on $A\otimes A\otimes A$.

We call $A$ with a comultiplication $\Delta$ a multiplier Hopf algebra if the linear maps $T_{1},T_{2}$ on $A\otimes A$ are bijective. Moreover, we call $A$ regular if $\tau\Delta$, where $\tau$ is the flip, is again a comultiplication such that $(A,\tau\Delta)$ is also a multiplier Hopf algebra. If $A$ is a $*$-algebra, we call $\Delta$ a comultiplication if it is also a $*$-homomorphism. A multiplier Hopf $*$-algebra is a $*$-algebra with a comultiplication, making it into a multiplier Hopf algebra.

An algebraic quantum group is a regular multiplier Hopf algebra with non-trivial invariant functionals (i.e., integrals).

Notice that we can use the Sweedler’s notation for $\Delta(x)$ when $x\in A$. The problem is that $\Delta(x)$ is not in $A\otimes A$ in general. By covering theory, we know however that $\Delta(x)(1\otimes y)\in A\otimes A$ for all $x,y\in A$. We can write (cf. [Dr-VD-Z, VD3]) $\Delta(x)=\sum x_{1}\otimes x_{2}$ for this. Now, we know that there is an element $e\in A$ such that $y=ey$ and we can think of $\Delta(x)=\sum x_{1}\otimes x_{2}$ to stand for $\Delta(x)(1\otimes e)$. Of course, this is still dependent on $y$. But we know that for several elements $y$, we can use the same $e$.

Let $A$ be a regular multiplier Hopf algebra (i.e. with a bijective antipode). By a left $A$-module $V$, we always mean a unital module (sometimes, we also say that $\rhd$ is a left action of $A$ on $V$). This means that $A\rhd V=V$. For all $v\in V$, we have an element $e\in A$ such that $e\rhd v=v$. Observe that for a unital algebra $A$, this condition is automatically satisfied. Similarly, a right $A$-module can be defined.

Let $Q$ denote a regular multiplier Hopf algebra and $A$ an associative algebra with or without identity. Assume that $\rhd$ is a left action of $Q$ on $A$ making $A$ into a left $Q$-module algebra, i.e., we have $x\rhd(ab)=\sum(x_{1}\rhd a)(x_{2}\rhd b)$ for all $x\in Q$ and $a,b\in A$. In this formula, $x_{1}$ is covered by $a$ (through the action) and $x_{2}$ is covered by $b$ (through the action). Similarly, assume that $\lhd$ is a right action of $Q$ on $A$ making $A$ into a right $Q$-module algebra, i.e., one has $(ab)\lhd x=\sum(a\lhd x_{1})(b\lhd x_{2})$, for all $x\in Q$ and $a,b\in A$. Then $A$ is called a $Q$-bimodule algebra, if it is a unital left $Q$-module and a unital right $Q$-module such that $(x\rhd a)\lhd y=x\rhd(a\lhd y)$, for all $x,y\in Q$ and $a\in A$.

Suppose that $\Gamma$ is a left coaction of $Q$ on $A$ making $A$ a left $Q$-comodule algebra. More precisely, $\Gamma:A\longrightarrow M(Q\otimes A)$ is an injective homomorphism so that for all $x\in Q$ and $a\in A$, we have $\Gamma(a)(x\otimes 1)$ and $(x\otimes 1)\Gamma(a)$ are in $Q\otimes A$. We use the Sweedler notation for these expressions, e.g. $\Gamma(a)(x\otimes 1)=\sum a_{(-1)}x\otimes a_{0}$. We say that the multiplier $\Gamma(a)$ is covered by $x\otimes 1$. Now the further requirement makes sense: $(\iota\otimes\Gamma)\Gamma(a)=(\Delta\otimes\iota)\Gamma(a)$ for all $a\in A$. In fact, the expression $(\iota\otimes\Gamma)\Gamma(a)=\sum a_{(-2)}\otimes a_{(-1)}\otimes a_{0}$ makes sense when it is understood that $\Delta(a_{(-1)})$ is replaced by $a_{(-2)}\otimes a_{(-1)}$. We have more or less the same rules for covering. We need to cover the factor $a_{(-1)}$ and possibly also $a_{(-2)}$ (and so on) by elements in $A$, left or right. In this case however, one cannot cover the factor $a_{0}$. Similarly, the right coaction $\Upsilon$ of $L$ on $A$ can be defined. Then $A$ is called a $Q$-$L$-bicomudule algebra if

[TABLE]

for all $x,y\in Q$ and $a\in A$.

Let $Q,L$ be multiplier Hopf algebras. Denote the category of all left $Q$-module algebras by ${}_{Q}{\cal MA}$, denote by the category of all right $L$-module algebras ${\cal MA}_{L}$, and denote by ${}_{Q}{\cal MA}_{L}$ the category of all $Q$-$L$-bimodule algebras. Similarly, denote the category of all left $Q$-comodule algebras by ${}^{Q}{\cal MA}$, denote by the category of all right $L$-module algebras ${\cal MA}^{L}$, and denote by ${}^{Q}{\cal MA}^{L}$ the category of all $Q$-$L$-bicomodule algebras.

Let $Q$ and $\widehat{Q}$ be a dual pair of algebraic quantum groups over $\mathbb{C}$. By Pontryagin duality, identifying $\widehat{\widehat{Q}}=Q$, we have a natural dual pairing $Q\otimes\widehat{Q}\longrightarrow\mathbb{C}$ is written as

[TABLE]

Note that the antipode of $Q^{op}$ and $Q^{cop}$ is given by $S^{-1}$ and the antipode of $Q^{op,cop}$ by $S$. Also, $\widehat{Q^{op}}=\widehat{Q}^{cop}$, $\widehat{Q^{cop}}=\widehat{Q}^{op}$ and $\widehat{Q^{op,cop}}=\widehat{Q}^{cop,op}$.

Obviously, after a permutation of tensor factors $A\otimes Q\longleftrightarrow Q\otimes A$ a left coaction of $Q$ may always be viewed as a right coaction by $Q^{cop}$ and vice versa.

Next, we recall that there is a one-to-one correspondence between right (left) coactions of $Q$ on $V$ and left (right) actions, respectively, of $\widehat{Q}$ on $V$ given for $\widehat{x}\in\widehat{Q}$ and $v\in V$ by

[TABLE]

and

[TABLE]

As a particular example we recall the case $V=Q$ with $\Gamma=\Upsilon=\Delta$. In this case we denote the associated left and right actions of $\widehat{x}\in\widehat{Q}$ on $y\in Q$ by $\widehat{x}\blacktriangleright y$ and $y\blacktriangleleft\widehat{x}$, respectively, see [De].

We refer to [VD1-VD3] for the theory of multiplier Hopf algebras. For the use of the Sweedler notation in this setting, we refer to [Dr-VD] and [Dr-VD-Z]. For pairings of multiplier Hopf algebras, the main reference is [Dr-VD]. An important reference for this paper is of course [P-O] where the theory of a more general version of the so-called L-R-smash product for (quasi-)Hopf algebras is developed. Finally, for some related constructions with multiplier Hopf algebras, see [De, De-VD-W, De-VD] and [VD-VK, W, W-L].

2. Twisted tensor products

In this section, we will study the notions of a generalized twisted tensor product, a generalized twisted smash product and a generalized L-R-smash product. We also study some isomorphisms between them.

2.1 Generalized twisted tensor products

Let $A,B$ be algebras and suppose that there are two $k$-linear maps $R:B\otimes A\longrightarrow A\otimes B$ and $T:A\otimes B\longrightarrow A\otimes B$ such that for $a,c\in A,b,d\in B$

[TABLE]

Here we write $R(b\otimes a)=a_{R}\otimes b_{R}=a_{r}\otimes b_{r}$ and $T(a\otimes b)=a_{T}\otimes b_{T}=a_{t}\otimes b_{t}$ for all $a\in A$ and $b\in B$, and $R_{12}=(R\otimes\iota)\in Hom_{k}(B\otimes A\otimes X,A\otimes B\otimes X)$, $R_{23}=(\iota\otimes R)\in Hom_{k}(X\otimes B\otimes A,X\otimes A\otimes B)$, $T_{12}=(T\otimes\iota)\in Hom_{k}(A\otimes B\otimes X,A\otimes B\otimes X)$, $T_{13}\in Hom_{k}(A\otimes X\otimes B,A\otimes X\otimes B)$ works on the first and the third componets as $T$ and $T_{23}=(\iota\otimes T)\in Hom_{k}(X\otimes A\otimes B,X\otimes A\otimes B)$ for any $k$-vector space $X$.

One can define a generalized L-R-twisted tensor product $A\#_{RT}B=A\otimes B$ as spaces with a new multiplication given by the formula:

[TABLE]

for all $a,c\in A$ and $b,d\in B$. Here, $T_{14}\in End_{k}(A\otimes B\otimes A\otimes B)$ works on the first and the forth componets as $T$.

Proposition 2.1.1. The above multiplication on $A\#_{RT}B=A\otimes B$ is associative.

Proof. For all $a,c,x\in A$ and $b,d,y\in B$. We compute

[TABLE]

and

[TABLE]

This completes the proof. $\blacksquare$

Remark 2.1.2. When $T$ is an identity map, then a generalized L-R-twisted tensor product $A\#_{RT}B=A\otimes B$ becomes a twisted tensor product $A\#_{R}B=A\otimes B$. As a space, $A\#_{R}B=A\otimes B$ with the following product defined by the formula:

[TABLE]

for all $a,c\in A$ and $b,d\in B$. Here, $R_{23}=\iota_{A}\otimes R\otimes\iota_{B}:A\otimes B\otimes A\otimes B\longrightarrow A\otimes A\otimes B\otimes B$.

If $A$ and and $B$ have a unit $1$, we suppose that $R$ satisfies $R(1\otimes a)=a\otimes 1$ and $R(b\otimes 1)=1\otimes b$ for all $a\in A$ and $b\in B$. Then $1\otimes 1$ is the unit of $A\#_{R}B$. Furthermore, the maps $A\longrightarrow A\#_{R}B:a\mapsto a\otimes 1$ and $B\longrightarrow A\#_{R}B:b\mapsto 1\otimes b$ are algebra embeddings. For more details on the above results, we refer to [VD-VK].

Definition 2.1.3. Let $A$ and $B$ be two algebras without unit, but with non-degenerate products. Let $R:B\otimes A\longrightarrow A\otimes B$ and $T:A\otimes B\longrightarrow A\otimes B$ be two $k$-linear maps. Then one can define the following two $k$-linear maps $R*T$ and $T*R$ from $A\otimes B\otimes A\otimes B$ to $A\otimes B$:

[TABLE]

for all $a,b\in A$ and $x,y\in B$.

Proposition 2.1.4. Let $A$ and $B$ be two algebras without unit, but with non-degenerate products. If $R$ and $T$ are bijective such that (1): $(T*R)(x\otimes y\otimes a\otimes b)=0$ implies that $x\otimes y=0$ and (2): $(R*T)(a\otimes b\otimes x\otimes y)=0$ implies that $x\otimes y=0$ for all $a,x\in A$ and $b,y\in B$, then the product in $A\#_{RT}B$ is non-degenerate.

Proof. Suppose that there is an element $\sum a_{i}\#b_{i}\in A\#_{RT}B$ such that $(\sum a_{i}\#b_{i})$ $(a\#b)=0$ for all $a\in A,b\in B$. Then we have that

[TABLE]

this implies (by Eq.(2.2)) that

[TABLE]

and by Eq.(2.3) and Eq.(2.5), one has that

[TABLE]

By the assumption (1), we have

[TABLE]

Thus, we obtain that $\sum a_{i}\#b_{i}=0$.

Similarly, by Eq.(2.1), (2.4), (2.5) and the assumption (2), one can show that $(a\#b)(\sum a_{i}\#b_{i})=0$ for all $a\in A,b\in B$ implies that $\sum a_{i}\#b_{i}=0$. $\blacksquare$

Corollary 2.1.5. Let $A$ and $B$ be two algebras without unit, but with non-degenerate products.

(1) ([De, Proposition 1.1]) If $R$ is bijective, then the product in $A\#_{R}B$ is non-degenerate.

(2) If $T$ is bijective, then the product in $A\#_{T}B$ is non-degenerate.

Proof. (1) In this case, $T=id_{A\otimes B}$. Thus the conditions (1): $(x_{T}a)_{R}\otimes(y_{R}b)_{T}=0$ implies that $x\otimes y=0$ and (2): $(ax_{R})_{T}\otimes(by_{T})_{R}=0$ implies that $x\otimes y=0$ for all $a,x\in A$ and $b,y\in B$, become that (1): $(xa)_{R}\otimes y_{R}b=0$ implies that $x\otimes y=0$ and (2): $ax_{R}\otimes(by)_{R}=0$ implies that $x\otimes y=0$ for all $a,x\in A$ and $b,y\in B$. We now explain how these two conditions hold. In fact, for the condition (1), since the product on $B$ is non-degenerate, $(xa)_{R}\otimes y_{R}b=0$ implies that $(xa)_{R}\otimes y_{R}=0$ and thus $x\otimes y=0$ since the product on $B$ is non-degenerate. Similarly, the condition (2) holds too.

(2) In this case, $R=id_{A\otimes B}$. The check is similar to the one in the part (1). $\blacksquare$

Theorem 2.1.6. Let $A$ and $B$ be Hopf algebras with two $k$-linear maps $R:B\otimes A\longrightarrow A\otimes B$ and $T:A\otimes B\longrightarrow A\otimes B$ such that Eq.(2.1)-(2.5) hold and

[TABLE]

Here we write $R(b\otimes a)=a_{R}\otimes b_{R}=a_{r}\otimes b_{r}$ and $T(a\otimes b)=a_{T}\otimes b_{T}=a_{t}\otimes b_{t}$ for all $a\in A$ and $b\in B$.

Then $(A\#_{RT}B,\Delta_{RT},\varepsilon_{RT},S_{RT})$ is a bialgebra, where the comultiplication, $\Delta_{RT}$ and $\varepsilon_{RT}$ are given as

[TABLE]

for all $a,c\in A$ and $b,d\in B$.

Furthermore, if

[TABLE]

then $(A\#_{RT}B,\Delta_{RT},\varepsilon_{RT},S_{RT})$ is a Hopf algebra with $S_{RT}$ given by

[TABLE]

Proof. From Proposition 2.1.1, we know that $A\#_{RT}B$ is an algebra with unity $1_{A}\#1_{B}$. Clearly $\Delta_{RT}$ is coassociative and $\varepsilon_{RT}$ is counity.

Take $a,c\in A$ and $b,d\in B$. We now prove that $\Delta_{RT}$ is a homomorphism.

[TABLE]

From the counitary property (2.8) we obtain that $\varepsilon_{RT}$ is a homomorphism on $A\#_{RT}B$. Therefore, $(A\#_{RT}B,\Delta_{RT},\varepsilon_{RT},S_{RT})$ is a bialgebra.

We show that $m_{A\#_{RT}B}(S_{RT}\otimes id_{A\otimes B})(\Delta_{RT}(a\#b))=\varepsilon_{RT}(a\#b)(1_{A}\#1_{B})$.

[TABLE]

Similarly, by Eq.(2.1), (2.4), (2.5) and Eq.(2.12), we can get

[TABLE]

Thus, $(A\#_{RT}B,\Delta_{RT},\varepsilon_{RT},S_{RT})$ is a Hopf algebra. $\blacksquare$

Remark 2.1.7. In the $(A\#_{RT}B,\Delta_{RT},\varepsilon_{RT},S_{RT})$. We have

(1) when $T$ is the identity map, i.e, $T=id_{A\otimes B}$, then the conditions (2.11) and (2.12) hold. In fact, we have

[TABLE]

and so Eq.(2.11) is obtained. Similarly,

[TABLE]

In this case, we have already obtained the result of [De, Theorem 2.1].

(2) when $R$ is the flip map, i.e., $\tau$, then the conditions (2.11) and (2.12) hold too.

(3) When $R$ and $T$ are not trivial, we will give the example (see Example 2.3.1 (3)).

Proposition 2.1.8. Let $A$ and $B$ be Hopf $*$-algebras with two $k$-linear maps $R:B\otimes A\longrightarrow A\otimes B$ and $T:A\otimes B\longrightarrow A\otimes B$ such that $R$ and $T$ satisfy all the conditions of Theorem 2.0.6. If furthermore

[TABLE]

then $(,\Delta_{RT},\varepsilon_{RT},S_{RT})$ is a Hopf $*$-algebra

Proof. By Theorem 2.1.6, one has that $(A\#_{RT}B,\Delta_{RT},\varepsilon_{RT},S_{RT})$ is a Hopf algebra. By the conditions that $R(*_{B}\otimes*_{A})\tau$ and $T(*_{A}\otimes*_{B})$ are involutions on $A\otimes B$, we have that $A\#_{RT}B$ is a $*$-algebra when $(a\#b)^{*}=TR(b^{*}\otimes a^{*})$. In what follows, we check that $\Delta_{RT}$ is a $*$-homomorphism on $A\#B$.

[TABLE]

This finishes the proof. $\blacksquare$

Proposition 2.1.9. Let $m\in M(A),n\in M(B)$, $M\in M(A\otimes A)$ and $N\in M(B\otimes B)$ be multipliers. Then

(1) we have that $m\#n\in M(A\#_{RT}B)$ where $m\#n$ is defined by

[TABLE]

for all $a\in A$ and $x\in B$.

(2) One has that $M\#N\in M((A\#_{RT}B)\otimes(A\#_{RT}B))$ where $M\#N$ is defined as follows:

[TABLE]

for all $a,b\in A$ and $x,y\in B$, where we write $M_{1}=M^{1}_{1}\otimes M^{2}_{1}$ and $N_{1}=N^{1}_{1}\otimes N^{2}_{1}$.

Proof. (1) For all $a,b\in A$ and $x,y\in B$.

[TABLE]

(2) Similarly.

This finishes the proof. $\blacksquare$

Definition 2.1.10. Let $A$ and $B$ be multiplier Hopf algebras with two $k$-linear maps $R:B\otimes A\longrightarrow A\otimes B$ and $T:A\otimes B\longrightarrow A\otimes B$ as described above. One defines:

[TABLE]

and

[TABLE]

We now prove that these candidates $T_{1}^{A\#_{RT}B}$ and $T_{2}^{A\#_{RT}B}$ defined by $R*T$ and $T*R$, respectively can be used to define a good comultiplication.

Proposition 2.1.11. Take the notations as above. For all $a,b,c\in A$ and $x,y,z\in B$, define

[TABLE]

and

[TABLE]

Then $\Delta_{RT}(a\#x)$ is a two-sided multiplier of $(A\#_{RT}B)\otimes(A\#_{RT}B)$, where $\Delta_{RT}(a\#x)$ $=$ $\Delta(a)\#\Delta(x)$. Furthermore, $\Delta_{RT}$ is coassociative on $A\#_{RT}B$.

Proof. For all $a,b\in A$ and $x,y\in B$, we compute.

[TABLE]

Similarly, by Eq.(2.2), (2.3) and (2.5) we have

[TABLE]

Firstly, for $a\in A,x\in B$, we prove that $\Delta_{RT}(a\#x)$ is a two-sided multiplier of $(A\#_{RT}B)\otimes(A\#_{RT}B)$. In fact, notice that $\Delta_{RT}(a\#x)=\Delta_{A}(a)\#\Delta_{B}(x)$ and recall from Proposition 2.1.8 that for $\Delta_{A}(a)\in M(A\otimes A)$ and $\Delta_{B}(x)\in M(B\otimes B)$, we can form the multiplier $\Delta_{A}(a)\#\Delta_{B}(x)\in M((A\#_{RT}B)\otimes(A\#_{RT}B))$. We can easily check that $\Delta_{RT}(a\#x)=\Delta_{A}(a)\#\Delta_{B}(x)$. It is clear that in the case that $A$ and $B$ are Hopf algebras, $\Delta_{RT}(a\#x)=(id_{A}\otimes\tau\otimes id_{B})(\Delta_{A}(a)\otimes\Delta_{B}(x))$.

Then in order to check that $\Delta_{RT}$ is coassociative on $A\#_{RT}B$, we have to prove that

[TABLE]

This calculations are straightforward by making use of the expressions

[TABLE]

and

[TABLE]

and the coassociativity of $\Delta_{A}$ and $\Delta_{B}$.

This completes the proof. $\blacksquare$

We now have the main result of this section as follows.

Theorem 2.1.12. Let $A$ and $B$ be multiplier Hopf algebras with two $k$-linear maps $R:B\otimes A\longrightarrow A\otimes B$ and $T:A\otimes B\longrightarrow A\otimes B$ such that Eq.(2.1)-(2.5) hold. If $R$ and $T$ are bijective such that (1): $(T*R)(x\otimes y\otimes a\otimes b)=0$ implies that $x\otimes y=0$ and (2): $(R*T)(a\otimes b\otimes x\otimes y)=0$ implies that $x\otimes y=0$ for all $a,x\in A$ and $b,y\in B$, and

[TABLE]

then $(A\#_{RT}B,\Delta_{RT},\varepsilon_{RT},S_{RT})$ is a multiplier Hopf algebra, where the multiplication, $\Delta_{RT}$, $\varepsilon_{RT}$ and $S_{RT}$ are given as

[TABLE]

for all $a,c\in A$ and $b,d\in B$.

Proof. We will finish the proof with the following steps:

(1) From the conditions Eq.(2.1)-(2.5) and Proposition 2.1.4, and the bijectivities of $T$ and $R$ we have that $A\#_{RT}B$ is an associative algebra with non-degenerate product.

(2) From Proposition 2.1.10 we conclude that $\Delta_{RT}$ is coassociative on $A\#_{RT}B$.

(3) We prove that $\Delta_{RT}:(A\#_{RT}B)\longrightarrow M((A\#_{RT}B)\otimes(A\#_{RT}B))$ is a homomorphism. For all $a,b\in A$ and $x,y\in B$, we have

[TABLE]

(4) Define the counit $\varepsilon_{RT}$ on $A\#_{RT}B$ by $\varepsilon_{RT}=\varepsilon_{A}\otimes\varepsilon_{B}$. We have to prove that for all $a,b\in A$ and $x,y\in B$

(i) $(\varepsilon_{RT}\otimes id_{A}\otimes id_{B})\Delta_{RT}(a\#x)((1\#1)\otimes(b\#y))=(a\#x)(b\#y)$;

(ii) $(id_{A}\otimes id_{B}\otimes\varepsilon_{RT})((a\#x)\otimes(1\#1))\Delta_{RT}(b\#y)=(a\#x)(b\#y)$.

We prove (i), the proof of (ii) is similar. From Proposition 2.1.10 we have that for $a,b\in A$ and $x,y\in B$

[TABLE]

where $T^{-1}(b\otimes y)=\sum_{i}y_{i}\otimes b_{i}$ and $R^{-1}(\sum_{i}y_{i}\otimes b_{i})=\sum_{ij}b_{i}{}^{j}\otimes y_{i}{}^{j}$.

So,

[TABLE]

(5) Because $T_{1}^{A\#_{RT}B}$ is surjective and $\Delta_{RT}$ is a homomorphism, $\varepsilon_{RT}$ is a homomorphism. This can be proved in a similar way as in [VD1, Lemma 3.5].

(6) On $A\#_{RT}B$, define the antipode $S_{RT}=T\circ R\circ(S_{B}\otimes S_{A})\tau$ which is an invertible map. We have to prove that for all $a,b\in A$ and $x,y\in B$

(i) $m_{A\#_{RT}B}(S_{RT}\otimes id_{A}\otimes id_{B})\Delta_{RT}(a\#x)((1\#1)\otimes(b\#y))=\varepsilon_{RT}(a\#x)(b\#y)$;

(ii) $m_{A\#_{RT}B}(id_{A}\otimes id_{B}\otimes S_{RT})((a\#x)\otimes(1\#1))\Delta_{RT}(b\#y)=(a\#x)\varepsilon_{RT}(b\#y)$.

We prove (i), the proof of (ii) is similar. In fact, start again from the following equation:

[TABLE]

where $T^{-1}(b\otimes y)=\sum_{i}y_{i}\otimes b_{i}$ and $R^{-1}(\sum_{i}y_{i}\otimes b_{i})=\sum_{ij}b_{i}{}^{j}\otimes y_{i}{}^{j}$.

Then we have

[TABLE]

(7) Because $T_{1}^{A\#_{RT}B}$ is surjective and $\Delta_{RT}$ is a homomorphism, $S_{RT}$ is a anti-homomorphism. The proof is similar to the proof of [VD1, Lemma 4.4].

It follows from [VD2, Proposition 2.9] that we now conclude that $(A\#_{RT}B,\Delta_{RT},$ $\varepsilon_{RT},S_{RT})$ is a (regular) multiplier Hopf algebra. $\blacksquare$

Proposition 2.1.13. Let $A$ and $B$ be multiplier Hopf $*$-algebras with two $k$-linear maps $R:B\otimes A\longrightarrow A\otimes B$ and $T:A\otimes B\longrightarrow A\otimes B$ such that $R$ and $T$ satisfy all the conditions of Theorem 2.1.12. If furthermore

[TABLE]

then $(A\#_{RT}B,\Delta_{RT},\varepsilon_{RT},S_{RT})$ is a multiplier Hopf $*$-algebra.

Proof. Straightforward. $\blacksquare$

Proposition 2.1.14. Let $A$ and $B$ be multiplier Hopf algebras as in Theorem 2.1.12. Let $\psi_{A}$ (resp. $\psi_{B}$) be a right integral on $A$ (resp. $B$). Then $\psi_{A}\otimes\psi_{B}$ is a right integral on the multiplier Hopf algebra $(A\#_{RT}B,\Delta_{RT},\varepsilon_{RT},S_{RT})$.

Proof. For all $a,b\in A$ and $x,y\in B$,

[TABLE]

This finishes the proof. $\blacksquare$

Proposition 2.1.15. Let $A$ and $B$ be multiplier Hopf algebras as in Theorem 2.1.12. Let $\varphi_{A}$ (resp. $\varphi_{B}$) be a left integral on $A$ (resp. $B$) with associated modular element $\delta_{A}$ (resp. $\delta_{B}$). Then the multiplier $\delta_{A}\#\delta_{B}$ is the modular element in $M(A\#_{RT}B)$ associated to $\varphi_{A}\otimes\varphi_{B}$.

Proof. Recall from [VD2] that the modular element $\delta_{A}$ in $M(A)$ for a multiplier Hopf algebra $A$ is given by $(\varphi_{A}\otimes id)\Delta(a)$ when $\varphi_{A}(a)=1$. Now, for the multiplier Hopf algebra $(A\#_{RT}B,\Delta_{RT},\varepsilon_{RT},S_{RT})$, we have that modular element $\delta_{A\#_{RT}B}$ associated to $\varphi_{A}\otimes\varphi_{B}$ is given as

[TABLE]

We claim that $\delta_{A\#_{RT}B}=\varphi_{A}\otimes\varphi_{B}$

For all $a,b\in A$ and $x,y\in B$, we have

[TABLE]

This completes the proof. $\blacksquare$

2.2 Generalized smash products

Let $Q$ be a regular multiplier Hopf algebra and let $A$ be a right $Q$-module algebra and $B$ a right $Q$-comodule algebra. Then there are a $k$-linear maps $R:A\otimes B\longrightarrow B\otimes A,a\otimes b\mapsto\sum b_{0}\otimes(a\lhd b_{(1)})$ for $a\in A$ and $b\in B$. By Remark 2.1.2, we may define a right smash product $B\#_{r}^{Q}A$ as $k$-vector space with multiplication

[TABLE]

for $a,a^{\prime}\in A,b,b^{\prime}\in B$, where we use the notation $b\#a$ in place of $b\otimes a$ to emphasize the new algebraic structure.

Let $Q$ be a regular multiplier Hopf algebra and let $A$ be a left $Q$-module algebra and $B$ a left $Q$-comodule algebra. Then there are another $k$-linear maps $R:B\otimes A\longrightarrow A\otimes B,b\otimes a\mapsto\sum b_{(-1)}\rhd a\otimes b_{0}$ for $a\in A$ and $b\in B$. Similarly, by Remark 2.1.2, $A\#_{l}^{Q}B$ denotes the associative algebra structure on $A\otimes B$ given by

[TABLE]

for $a,a^{\prime}\in A,b,b^{\prime}\in B$, containing $A$ and $B$ as subalgebras.

We have obvious embeddings of $A$ and $B$ in the multiplier algebra of $B\#_{r}^{Q}A$. And we note that $B\#_{r}^{Q}A=BA=AB$ with $ab=\sum b_{0}(a\lhd b_{(1)})$ for $a\in A,b\in B$. Similar statements hold in $A\#_{l}^{Q}B$.

Example 2.2.1 (1) Given a left coaction $\Gamma:A\longrightarrow M(Q\otimes A)$ of $Q$ on $A$ with dual right $\widehat{Q}$-action $\lhd$. One has the right smash product $\widehat{Q}\#_{r}^{\widehat{Q}}A$ to be the vector space $\widehat{Q}\otimes A$ with associative algebra structure given for $\widehat{x},\widehat{y}\in\widehat{Q}$ and $a,b\in A$ by

[TABLE]

(2) Similarly if $\Upsilon:A\longrightarrow M(A\otimes Q)$ is a right coaction of $Q$ on $A$ with dual left $\widehat{Q}$-action $\rhd$, then the multiplication of the algebra $A\#_{l}^{\widehat{Q}}\widehat{Q}$ is given by

[TABLE]

for $\widehat{x},\widehat{y}\in\widehat{Q}$ and $a,b\in A$

We remark that any right smash product $\widehat{Q}\#_{r}^{\widehat{Q}}A$ can be identified with an associated left smash product $A\#_{l}^{\widehat{Q}^{cop}}\widehat{Q}^{cop}$, where $\Upsilon:A\longrightarrow M(A\otimes Q^{op})$ is the right coaction given by

[TABLE]

with $\tau_{Q,A}:Q\otimes A\longrightarrow A\otimes Q$ being the permutation of tensor factors. In fact we have

Lemma 2.2.2. Let $\Gamma:A\longrightarrow M(Q\otimes A)$ and $\Upsilon:A\longrightarrow M(A\otimes Q^{op})$ be a pair of left and right coactions, respectively, related by (2.2.3). Then we have

[TABLE]

Proof. Let $\rhd:\widehat{Q}^{cop}\otimes A\longrightarrow A$ and $\lhd:A\otimes\widehat{Q}\longrightarrow A$ be the left and right actions dual to $\Gamma$ and $\Upsilon$, respectively. The equation (2.2.3) is equivalent to $\widehat{x}\rhd a=a\lhd\widehat{S^{-1}}(x)$ for all $\widehat{x}\in\widehat{Q}$ and $a\in A$.

Define a linear map $f:A\#_{l}^{\widehat{Q}^{cop}}\widehat{Q}^{cop}\longrightarrow\widehat{Q}\#_{r}^{\widehat{Q}}A$ as

[TABLE]

It is easy to check that its inverse is given by

[TABLE]

for $\widehat{x}\in\widehat{Q}$ and $a\in A$.

The remaining thing is straightforward. $\blacksquare$

Let $Q,L$ be Hopf algebras and let $A$ be a right $Q$-module algebra and $B$ a right $L$-comodule algebra. Let $f:L\longrightarrow Q$ be a bialgebra map. Then there are a $k$-linear maps $R:A\otimes B\longrightarrow B\otimes A,a\otimes b\mapsto\sum b_{0}\#(a\lhd f(b_{(1)}))$ for $a\in A$ and $b\in B$. By Remark 2.1.2, we can define a right smash product $B\#_{r}^{(Q,L,f)}A$ as $k$-vector space with multiplication

[TABLE]

for $a,a^{\prime}\in A,b,b^{\prime}\in B$, where we use the notation $b\#a$ in place of $b\otimes a$ to emphasize the new algebraic structure.

Then the proof of the following proposition is straightforward.

Proposition 2.2.3. With the above notation. $B\#_{r}^{(Q,L,f)}A$ is an associative algebra.

Similarly, let $A$ be a left $Q$-module algebra and $B$ a left $L$-comodule algebra. Let $f:L\longrightarrow Q$ be a bialgebra map. Then there are a $k$-linear maps $R:B\otimes A\longrightarrow A\otimes B,b\otimes a\mapsto\sum(f(b_{(-1)})\rhd a)\#b_{0}$ for $a\in A$ and $b\in B$. By Remark 2.1.2, $A\#_{l}^{(Q,L,f)}B$ denotes the associative algebra structure on $A\otimes B$ given by

[TABLE]

for $a,a^{\prime}\in A,b,b^{\prime}\in B$, containing $A$ and $B$ as subalgebras.

2.3 Generalized twisted smash products

Let $Q$ be a regular multiplier Hopf algebra. Let $A$ be a $Q$-bimodule algebra, i.e, $A\in{}_{Q}{\cal MA}_{Q}$ and let $B$ be a $Q$-bicomodule algebra, i.e., $B\in{}^{Q}{\cal MA}^{Q}$. In this section, we consider an algebra $P$ that is a generalized smash product of $A$ and $B$. The construction has probably been studied in [W] for Hopf algebras but not yet for multiplier Hopf algebras. However, the results and the arguments are very similar to the theory of smash products as developed in [Dr-VD-Z]. Therefore, in the following proposition, we do not give all the details. We concentrate on the correct statements and briefly indicate how things are proven.

Define $B\star^{Q}_{r}A=B\otimes A$ as $k$-vector space with multiplication

[TABLE]

for $a,a^{\prime}\in A,b,b^{\prime}\in B$.

Similarly $A\star_{l}^{Q}B$ denotes the associative algebra structure on $A\otimes B$ given by

[TABLE]

for $a,a^{\prime}\in A,b,b^{\prime}\in B$.

Proposition 2.3.1 $B\star^{Q}_{r}A$ and $A\star_{l}^{Q}B$ as above are associative algebras.

Proof. For $B\star^{Q}_{r}A$, the twist map is given by the formula: $R:A\otimes B\longrightarrow B\otimes A,a\otimes b=\sum b_{0}\otimes(S^{-1}(b_{(-1)})\rhd a\lhd b_{(1)})$ for $a\in A$ and $b\in B$. By Remark 2.1.2, $B\star^{Q}_{r}A$ is an associative algebra.

Similarly for $A\star_{l}^{Q}B$ and the proof is finished. $\blacksquare$

The algebra $B\star^{Q}_{r}A$ (resp. $A\star_{l}^{Q}B$) is called a generalized right (resp. left) smash product, which is denoted by $P$ (resp. $U$). Just as in the case of smash products, we have obvious embeddings of $A$ and $B$ in the multiplier algebra of $P$ (resp. $U$) and if we identify these two algebras with their images in $M(P)$ (resp. $M(U)$), we see that $P$ (resp. $U$) is the linear span of elements $ab$ (resp. $ba$) with $a\in A$ and $b\in$ and that we have the commutation rules

i) $A$ and $B$ commute,

ii) $ab=\sum b_{0}(S^{-1}(b_{(-1)})\rhd a\lhd b_{(1)})$

   (resp. $ba=\sum(b_{(-1)}\rhd a\lhd S^{-1}(b_{(1)}))b_{0}$), for $a\in A,b\in B$.

Therefore we can view $P$ (resp. $U$)as the algebra generated by $A$ and $B$ subject to these commutation rules.

Example 2.3.2. (1) This construction reduces to well-know constructions in the following three special situations. If the multiplier Hopf algebra $Q$ is trivial, then we obtain for $P$ simply the tensor product algebra $A\otimes B$. If the left action and coaction of $Q$ on $A$ and $B$ are trivial, respectively, we obtain the right smash product $B\#^{Q}_{r}A$.

(2) Let $Q$ denote an algebraic quantum group. Then $\widehat{Q}$ is the $Q$-bimodule algebra. Let $B$ be a $Q$-bicomodule algebra. Then we have two generalized twisted smash products $B\star^{Q}_{r}\widehat{Q}$ and $\widehat{Q}\star^{Q}_{l}B$.

(3) Let $Q$ denote a regular multiplier Hopf algebra with antipode $S^{2}=\iota$ and $A$ a $Q$-bimodule algebra. Then the generalized twisted smash product $A\star_{l}^{Q}Q$ is isomorphic to an ordinary smash product $A\#Q$, where the left $Q$-action on $Q$ is now given by $x\rightarrow a=\sum x_{1}\rhd aS(x_{2})$, for all $a\in A$ and $x\in Q$. Recall that in the original paper [Dr-V-Z], one developed the theory for left actions.

2.4 Generalized L-R-smash products

The L-R-smash product was introduced and studied in a series of papers [P-O], with motivation and examples coming from the theory of deformation quantization. However, we will study the case slightly different from [P-O].

Let $Q$ be a multiplier Hopf algebra, $A$ a $Q$-bimodule algebra, i.e., $A\in{}_{Q}{\cal MA}_{Q}$ and $B$ a $Q$-bicomodule algebra, i.e., $B\in{}^{Q}{\cal MA}^{Q}$. Define $R:A\otimes B\longrightarrow B\otimes A$ and $T:B\otimes A\longrightarrow B\otimes A$ as $R(a\otimes b)=\sum b_{0}\otimes a\lhd b_{(1)}$ and $T(b\otimes a)=\sum b_{0}\otimes b_{(-1)}\rhd a$, respectively, for all $a\in A$ and $b\in B$. Then, by Eq. (2.1.1), we have the generalized L-R-twisted tensor product $B\diamondsuit^{Q}_{r}A$ with the multiplication by

[TABLE]

for $a,a^{\prime}\in A$ and $b,b^{\prime}\in B$.

Similarly, Define $R:B\otimes A\longrightarrow A\otimes B,b\otimes a\mapsto\sum(b_{(-1)}\rhd a)\otimes b_{0}$ and $T:A\otimes B\longrightarrow A\otimes B,a\otimes b\mapsto\sum(a\lhd b_{(1)})\otimes b_{0}$ for all $a\in A$ and $b\in B$. Then, by Eq. (2.1.1), we can define $A\diamondsuit^{Q}_{l}B=A\otimes B$ as $k$-vector space with multiplication

[TABLE]

for $a,c\in A,b,d\in B$.

Example 2.4.1. (1) Let $A$ be a right $Q$-module algebra. Then $A$ becomes an $Q$-bimodule algebra, with left $Q$-action given via $\varepsilon$. In this case the multiplication of $B\diamondsuit^{Q}_{r}A$ becomes

[TABLE]

for $a,a^{\prime}\in A,b,b^{\prime}\in B$, hence in this case $B\diamondsuit^{Q}_{r}A$ coincides with the generalized smash product $B\#^{Q}_{r}A$.

(2) We note that $Q$ itself is a $Q$-bicomodule algebra. So, in this case, the multiplication of $Q\diamondsuit^{Q}_{r}A$ specializes to

[TABLE]

for $a,b\in A,x,y\in Q$. If the left $Q$-action is trivial, then $Q\diamondsuit^{Q}_{r}A$ coincides with the smash product $Q\#^{Q}_{r}A$

(3) If we consider a usual Hopf algebra $Q$, and assume that $A$ is a $Q$-bimodule algebra. Define $R:Q\otimes A\longrightarrow A\otimes Q,q\otimes a\mapsto\sum q_{1}\rhd a\otimes q_{2}$ and $T:A\otimes Q\longrightarrow A\otimes Q,a\otimes q\mapsto\sum a\lhd q_{2}\otimes q_{1}$ for all $a\in A$ and $q\in Q$. Then, by Eq. (2.1.1), we can define $A\diamondsuit^{Q}_{l}Q=A\otimes Q$ as $k$-vector space with multiplication

[TABLE]

for $a,b\in A,q,p\in Q$. If $T(a\otimes q)=\sum a\lhd q_{1}\otimes q_{2}$ and $R(q\otimes a)=\sum q_{2}\rhd a\otimes q_{1}$, then the conditions (2.11) and (2.12) hold. In fact, we have

[TABLE]

and so Eq.(2.11) is obtained. Similarly, we have

[TABLE]

Proposition 2.4.2. $B\diamondsuit^{Q}A$ and $A\diamondsuit^{Q}_{l}B$ as above are associative algebras.

The proof of this result is straightforward.

Proposition 2.4.3. Let $A$ be a $Q$-bimodule algebra and let $B$ be a $Q$-bicomodule algebra. Then $B\diamondsuit^{Q}_{r}A\cong B\star^{Q}_{r}A$ and $A\diamondsuit^{Q}_{l}B\cong A\star^{Q}_{l}B$.

Proof. Define $\Phi:B\star^{Q}_{r}A\longrightarrow B\diamondsuit^{Q}_{r}A$ as

[TABLE]

for all $a\in A$ and $b\in B$. First, it is not hard to check that the map $\Phi$ has the inverse $\Psi:B\diamondsuit^{Q}_{r}A\longrightarrow B\star^{Q}_{r}A$ given by

[TABLE]

Then we check that $\Phi$ is a homomorphism as follows.

[TABLE]

for $a,a^{\prime}\in A,b,b^{\prime}\in B$.

Similarly for $A\diamondsuit^{Q}_{l}B$. This completes the proof. $\blacksquare$

Example 2.4.4. (1) Let $Q$ be a regular multiplier Hopf algebra and let $B$ be a $Q$-bicomodule algebra. If $A$ is a left $Q$-module algebra regarded as an $Q$-bimodule algebra with trivial right $Q$-action, then $A\star^{Q}_{l}B$ and $A\diamondsuit^{Q}_{l}B$ both coincide with $A\#^{Q}_{l}B$, and the isomorphism is just the identity.

(2) If $B=Q$, the maps $\Phi:B\star^{Q}_{r}A\longrightarrow B\diamondsuit^{Q}_{r}A$ and $\Psi:B\diamondsuit^{Q}_{r}A\longrightarrow B\star^{Q}_{r}A$ become:

[TABLE]

for all $a\in A$ and $b\in B$.

Let now $Q$ be a multiplier Hopf algebra, $A$ a $Q$-bimodule algebra and $B$ a $Q$-bicomodule algebra. Let $C$ be an algebra in the Yetter-Drinfeld category ${}^{Q}_{Q}{\cal YD}$ (see [De]) that is $C$ is both a left $Q$-module algebra and a left $Q$-comodule algebra. These two structures are crossed via the Yetter-Drinfeld compatibility condition in the following sense. For all $c\in C$ and $x,y\in Q$ we require

[TABLE]

Consider first the generalized smash product $A\#^{Q}_{l}C$, an associative algebra. From Eq.(2.4.4), it follows that $A\#^{Q}_{l}C$ becomes a $Q$-bimodule algebra, with $Q$-actions

[TABLE]

for all $x\in Q,a\in A$ and $c\in C$, hence we may consider the algebra $(A\#^{Q}_{l}C)\diamondsuit^{Q}_{l}B$. Similarly, for $B\diamondsuit^{Q}_{r}(A\#^{Q}_{l}C)$.

Meanwhile, consider the generalized smash product $C\#^{Q}_{l}B$, an associative algebra. Using the condition Eq.(2.4.4), one can see that $C\#^{Q}_{l}B$ becomes a $Q$-bicomodule algebra, with $Q$-coactions:

[TABLE]

for all $a\in A$ and $b\in B$, hence we may consider the algebra $A\diamondsuit^{Q}_{l}(C\#^{Q}_{l}B)$.

Proposition 2.4.5. We have an algebra isomorphism $(A\#^{Q}_{l}C)\diamondsuit^{Q}_{l}B\cong A\diamondsuit^{Q}_{l}(C\#^{Q}_{l}B)$, given by the trivial identification.

Proof. We compute the multiplication in $(A\#^{Q}_{l}C)\diamondsuit^{Q}_{l}B$ as follows:

[TABLE]

The multiplication in $A\diamondsuit^{Q}_{l}(C\#^{Q}_{l}B)$ is:

[TABLE]

Hence the two multiplications coincide, completing the proof. $\blacksquare$

Since the L-R-smash product coincides with the generalized smash product if the right $Q$-action is trivial, we also obtain:

Corollary 2.4.6. If $Q,A,B$ are as above and $C$ is a left $Q$-module algebra, then we have an algebra isomorphism $(C\#^{Q}_{l}A)\#^{Q}_{l}B\cong C\#^{Q}_{l}(A\#^{Q}_{l}B)$, given by the trivial identification.

Let $Q$ be a regular multiplier Hopf algebra. Recall that the Drinfel’d double $D(Q)$ (generalizing the usual Drinfel’d double of a Hopf algebra) was introduced by Drabant and Van Daele in [Dr-VD] by a general procedure, and more explicit descriptions were obtained afterwards by Delvaux and Van Daele in [De-VD]. According to one of these descriptions, the algebra structure of $D(Q)$ is just the twisted smash product $\widehat{Q}\star Q^{cop}$. By transferring the whole structure of $D(Q)$ via the map $\Phi$, we can thus obtain a new realization of $D(Q)$, having the L-R-smash product $\widehat{Q}\diamondsuit Q^{cop}$ for the algebra structure.

Let $Q$ be a multiplier Hopf algebra. From [De-VD], an invertible element $R\in M(Q\otimes Q)$ is called a Drinfeld twist (or a gauge transformation) if

[TABLE]

If $R=R^{1}\otimes R^{2}\in M(Q\otimes Q)$ is a Drinfeld twist with inverse $R^{-1}=R^{-1}\otimes R^{-2}\in M(Q\otimes Q)$, then we can define a new multiplier Hopf algebra $Q_{R}$ (see, [Wa1]) with the same multiplication and counit as $Q$, for which the comultiplication and antipode are given by, for $x\in Q$

[TABLE]

where $T_{R}=R^{1}S(R^{2})$ is an invertible element of $M(Q)$ with the inverse $T_{R}^{-1}=S(R^{-1})R^{-2}$.

Remark: (1) It is easy to get that

[TABLE]

and

[TABLE]

(2) Let $x\in M(Q)$ be an invertible element such that $\varepsilon(x)=1_{Q}$. If $R$ is a twist for $Q$ then so is $R^{x}:=R\Delta(x)(x^{-1}\otimes x^{-2})$. The twists $R$ and $R^{x}$ are said to be gauge equivalent.

Let $Q$ be a multiplier Hopf algebra, $A$ a $Q$-bimodule algebra and $R\in M(Q\otimes Q)$ a Drinfeld twist. If we introduce on $A$ another multiplication, by $a\circ b=(R^{-1}\rhd a\lhd R^{1})(R^{-2}\rhd b\lhd R^{2})$ for all $a,b\in A$, and denote this structure by ${}_{R^{-1}}A_{R}$, then ${}_{R^{-1}}A_{R}$ is a $Q_{R}$-bimodule algebra, with the same $Q$-actions as $A$. We have

Proposition 2.4.6. Take the notations as above. Then ${}_{R^{-1}}A_{R}$ is a $Q_{R}$-bimodule algebra.

Proof. We first want to show that the product in ${}_{R^{-1}}A_{R}$ is associative and non-degenerate. Furthermore, $1_{Q}\in A$ remains the unit in $M({}_{R^{-1}}A_{R})$.

We compute

[TABLE]

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

Then, one has to show that ${}_{R^{-1}}A_{R}$ is a left $Q_{R}$-module algebra.

We note that $\Delta_{R}(x)=:\sum x_{[1]}\otimes x_{[2]}=\sum R^{1}x_{1}R^{-1}\otimes R^{2}x_{2}R^{-2}$

[TABLE]

for all $x\in Q,a,b\in A$

Similarly, one can check that ${}_{R^{-1}}A_{R}$ is a right $Q_{R}$-module algebra and that ${}_{R^{-1}}A_{R}$ is a $Q_{R}$-bimodule. This finishes the proof. $\blacksquare$

Suppose that we have a left $Q$-comodule algebra $B$; then on the algebra structure of $B$ one can introduce a left $Q_{R}$-comodule algebra structure (denoted by ${}^{R^{-1}}B$ in what follows) putting the same $Q$-coaction $\Gamma$ as $B$. Similarly, if $C$ is a right $Q$-comodule algebra, one can introduce on the algebra structure of $C$ a right $Q_{R}$-comodule algebra structure (denoted by $C^{R}$ in what follows) putting the same $Q$-coaction $\Gamma$ as $C$. One may check that if $B$ is a $Q$-bicomodule algebra, the left and right $Q_{R}$-comodule algebras ${}^{R^{-1}}B$ respectively $B^{R}$ actually define the structure of a $Q_{R}$-bicomodule algebra on $B$, denoted by ${}^{R^{-1}}B^{R}$.

Then the proof of the following result is straightforward.

Proposition 2.4.7. Take the notations as above. If $B$ is a $Q$-bicomodule algebra, then ${}^{R^{-1}}B^{R}$ is a $Q_{R}$-bicomodule algebra.

Proposition 2.4.8. Let $A$ be a $Q$-bimodule algebra and $B$ a $Q$-bicomodule algebra. Then we have two algebra isomorphisms:

[TABLE]

given by the trivial identification.

Proof. We only compute the multiplication in ${}^{R^{-1}}B^{R}\diamondsuit^{Q}_{r}{}_{R^{-1}}A_{R}$. Similar to ${}_{R^{-1}}A_{R}\diamondsuit^{Q}_{l}{}^{R^{-1}}B^{R}$.

[TABLE]

for $a,a^{\prime}\in A,b,b^{\prime}\in B$.

which is the multiplication of

This finishes the proof. $\blacksquare$

By Proposition 2.4.3, we have

Corollary 2.4.9. Take the notations and assumptions as above. We have

[TABLE]

3. Two-times twisted tensor products

In this section, we will study the notion of a two-times twisted tensor product.

3.1 Two-sided twisted tensor products

Let $A,B,C$ be algebras and let $R:C\otimes A\longrightarrow A\otimes C$ and $T:B\otimes C\longrightarrow C\otimes B$ be $k$-linear maps. Then we will write $R(c\otimes a)=a_{R}\otimes c_{R}=a_{r}\otimes c_{r}$ and $T(b\otimes c)=c_{T}\otimes b_{T}=c_{t}\otimes b_{t}$ for all $a\in A,b\in B$ and $c\in C$. One defines two-sided twisted tensor product $A\#_{R}C\#_{T}B=A\otimes C\otimes B$ as spaces with a new multiplication defined by the formula:

[TABLE]

or

[TABLE]

for all $a,a^{\prime}\in A,b,b^{\prime}\in B$ and $c,c^{\prime}\in C$, where $\sigma$ denotes the usual flip map on $B\otimes A$.

Let $Q,L$ be multiplier Hopf algebras. Let $A$ be in ${}_{Q}{\cal MA}$, $B\in{\cal MA}_{L}$, and let $C$ be in ${}^{Q}{\cal MA}^{L}$. If we define $R:C\otimes A\longrightarrow A\otimes C$ and $T:B\otimes C\longrightarrow C\otimes B$, respectively by $R(c\otimes a)=\sum c_{(-1)}\rhd a\otimes c_{0}$ and $T(b\otimes c)=c_{0}\otimes b\lhd c_{(1)}$ for all $a\in A,b\in B$ and $c\in C$, then we have a two-sided twisted tensor product $A\boxminus^{Q}_{l}C\boxminus^{L}_{r}B$ with the multiplication given by

[TABLE]

for $a,a^{\prime}\in A,b,b^{\prime}\in B,c,c^{\prime}\in C$, here we write $a\boxminus c\boxminus b$ for $a\otimes c\otimes b$.

Proposition 3.1.1. $A\boxminus^{Q}_{l}C\boxminus^{L}_{r}B$ defined above is an associative algebra.

Note that, given $A,B$ as above, $A\otimes B$ becomes a $Q$-bimodule algebra, with $Q$-actions

[TABLE]

for all $a\in A,x,y\in Q,b\in B$.

Then we have

Proposition 3.1.2. Let $A,B,C,Q$ be as above. Then we have the following algebra isomorphisms

(1) $\Phi:(A\otimes B)\diamondsuit^{L}_{l}C\cong A\boxminus^{Q}_{l}C\boxminus^{L}_{r}B,\quad\Phi((a\otimes b)\diamondsuit c)=a\boxminus c\boxminus b$;

(2) $\Psi:C\diamondsuit^{Q}_{r}(A\otimes B)\cong A\boxminus^{Q}_{l}C\boxminus^{Q}_{r}B,\quad\Psi(c\diamondsuit(a\otimes b))=a\boxminus c\boxminus b$;

(3) $(A\otimes B)\star^{L}_{l}C\cong(A\otimes B)\diamondsuit^{L}_{l}C\cong C\diamondsuit^{Q}_{r}(A\otimes B)\cong C\star^{Q}_{r}(A\otimes B)$;

(4) $(A\#^{Q}_{l}C)\#^{L}_{r}B=A\#^{Q}_{l}(C\#^{L}_{r}B)=A\boxminus^{Q}_{l}C\boxminus^{L}_{r}B$.

Proof. (1) We do calculations as follows:

[TABLE]

for $a,a^{\prime}\in A,b,b^{\prime}\in B,c,c^{\prime}\in C$.

(2) Similar to (1).

(3) follows (1), (2) and Proposition 2.4.3.

(4) We note that $A\#^{Q}_{l}C$ has an induced right $Q$-comodule coaction by the right coaction of $Q$ on $C$, i.e., $\Upsilon:A\#^{Q}_{l}C\longrightarrow A\#^{Q}_{l}C\otimes Q$, $\Upsilon(a\#c)=a\#c_{0}\otimes c_{(1)}$ for all $a\in A,c\in C$. Hence, $(A\#^{Q}_{l}C)\#^{Q}_{r}B=A\boxminus^{Q}_{l}C\boxminus^{Q}_{r}B$. Similarly, $A\#^{Q}_{l}(C\#^{Q}_{r}B)=A\boxminus^{Q}_{l}C\boxminus^{Q}_{r}B$.

This finishes the proof. $\blacksquare$

3.2 Two-sided L-R-smash products

Let $Q$ be a multiplier Hopf algebra, $A$ a right $Q$-comodule algebra, $B$ a left $Q$-comodule algebra and $C$ a $Q$-bimodule algebra. Define on $A\otimes C\otimes B$ a multiplication by the formula

[TABLE]

for $a,a^{\prime}\in A,b,b^{\prime}\in B,c,c^{\prime}\in C$.

Then this multiplication yields an associative algebra structure, denoted by $A\boxtimes^{Q}_{r}C\boxtimes^{Q}_{l}B$.

Note that, given $A,B$ as above, $A\otimes B$ becomes an $Q$-bicomodule algebra, with the following structure:

[TABLE]

and

[TABLE]

Proposition 3.2.1. If $Q,A,B,C$ are as above, then we have the following algebra isomorphisms:

(1) $\phi:(A\otimes B)\diamondsuit^{Q}_{r}C\cong A\boxtimes^{Q}_{r}C\boxtimes^{Q}_{l}B,\quad\phi((a\otimes b)\diamondsuit c)=a\boxtimes c\boxtimes b$;

(2) $\psi:C\diamondsuit^{Q}_{l}(A\otimes B)\cong A\boxtimes^{Q}_{r}C\boxtimes^{Q}_{l}B,\quad\psi(c\diamondsuit(a\otimes b))=a\boxtimes c\boxtimes b$;

(3) $(A\otimes B)\star^{Q}_{r}C\cong(A\otimes B)\diamondsuit^{Q}_{r}C\cong C\diamondsuit^{Q}_{l}(A\otimes B)\cong C\star^{Q}_{l}(A\otimes B)$;

(4) $(A\#^{Q}_{r}C)\#^{Q}_{l}B=A\#^{Q}_{r}(C\#^{Q}_{l}B)=A\boxtimes^{Q}_{r}C\boxtimes^{Q}_{l}B$.

Proof. (1) We compute:

[TABLE]

(2) and (3) Obvious.

(4) We note that $A\#^{Q}_{r}C$ has an induced $Q$-bimodule algebra by the one on $C$. This finishes the proof. $\blacksquare$

4. Twisted products

In this section we will introduce the notion of a Long module algebra and study the the twisted product of the given algebra.

Definition 4.1. Let $Q$ be a multiplier Hopf algebra.

(1) Let $A$ be a left $Q$-module algebra and a left $Q$-comodule algebra. $A$ is called a left-left $Q$-Long module algebra if the following condition holds:

[TABLE]

for all $x,y\in Q,a\in A$. The category of all left $Q$-Long module algebras is denoted by ${}^{Q}_{Q}{\cal LA}$.

(2) Let $A$ be a left $Q$-module algebra and a right $Q$-comodule algebra. $A$ is called a left-right $Q$-Long module algebra if the following condition holds:

[TABLE]

for all $x,y\in Q,a\in A$. The category of all left-right $Q$-Long module algebras is denoted by ${}_{Q}{\cal LA}^{Q}$.

(3) Let $A$ be a right $Q$-module algebra and a right $Q$-comodule algebra. $A$ is called a right-right $Q$-Long module algebra if the following condition holds:

[TABLE]

for all $x,y\in Q,a\in A$. The category of all right-right $Q$-Long module algebras is denoted by ${}{\cal LA}^{Q}_{Q}$.

(4) Let $A$ be a right $Q$-module algebra and a left $Q$-comodule algebra. $A$ is called a right-left $Q$-Long module algebra if the following condition holds:

[TABLE]

for all $x,y\in Q,a\in A$. The category of all right-left $Q$-Long module algebras is denoted by ${}^{Q}{\cal LA}_{Q}$.

Let $A\in{}^{Q}_{Q}{\cal LA}$. If we define a new multiplication on $A$, by

[TABLE]

then this multiplication defines a new algebra structure on $A$. The product $\bullet$ is called the left twisted product.

Example 4.2. Let $Q$ be a regular multiplier Hopf algebra with bijective antipode $S$, $A$ a $Q$-bimodule algebra and $B$ a $Q$-bicomodule algebra. Then $A$ becomes a left $Q\otimes Q^{op}$-module algebra with the following structure:

[TABLE]

and $B$ becomes a left $Q\otimes Q^{op}$-comodule algebra with the following structure

[TABLE]

It is easy to check that $A\otimes B$ is a left $Q\otimes Q^{op}$-Long module algebra with the natural structure. The corresponding twisted product on $A\otimes B$ is

[TABLE]

for all $a,a^{\prime}\in A$ and $b,b^{\prime}\in B$, and this is exactly the multiplication of the generalized right twisted smash product.

Similarly, for $A\in{\cal LA}^{Q}_{Q}$. If we define a new multiplication on $A$, by

[TABLE]

then this multiplication defines a new algebra structure on $A$. The product $\ast$ is called the right twisted product.

Let $Q$ be a regular multiplier Hopf algebra with bijective antipode $S$, let $A$ be a $Q$-bimodule algebra and a $Q$-bicomodule algebra such that $A\in{}^{Q}_{Q}{\cal LA}^{Q}_{Q}$, called four sides Long module category. Then we define a new multiplication on $A$ by

[TABLE]

and call it the L-R-twisted product.

Then it is easy to check that $A$ is a left $Q\otimes Q^{op}$-Long module algebra with the natural structure (see Example 4.2). The corresponding twisted product on $A$ is

[TABLE]

Proposition 4.3. $(A,\circledast)$ is an associative algebra.

Proof. We do calculations as follows:

[TABLE]

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

This finishes the proof. $\blacksquare$

Four sides Long module algebra category is in particular regarded as a left-left Long module algebra category, but in general the corresponding twisted products $\circledast$ and respectively $\bullet$ are different. On the other hand, any a left-left Long module algebra category can be regarded as a four sides Long module algebra category with trivial right action and coaction, and in this case the corresponding twisted products coincide.

Proposition 4.4. Let $A\in{}^{Q}_{Q}{\cal LA}^{Q}_{Q}$. With notation as before. Then the L-R-twisted product $\circledast$ can be obtained as a left twisting followed by a right twisting and also vice versa.

Proof. First consider the left twisted product algebra $(A,\bullet)$; it is easy to see that $(A,\bullet)\in{\cal LA}^{Q}_{Q}$, and the corresponding right twisted product becomes:

[TABLE]

for all $a,b\in A$.

Similarly, one can start with the right twisted product algebra $(A,\ast)$, for which $(A,\ast)\in{}^{Q}_{Q}{\cal LA}$, and the corresponding left twisted product coincides with the L-R-twisted product.

This finishes the proof. $\blacksquare$

Example 4.5. Let $Q$ be a regular multiplier Hopf algebra, $A$ a $Q$-bimodule algebra and $B$ a $Q$-bicomodule algebra. Take the algebra $A\otimes B$, which becomes a $Q$-bimodule algebra with actions $x\rhd(a\otimes b)=x\rhd a\otimes b$ and $(a\otimes b)\lhd x=a\lhd x\otimes b$, for all $x\in Q,a\in A,b\in B$, and a $Q$-bicomodule algebra, with coactions $\Gamma:A\otimes B\longrightarrow Q\otimes(A\otimes B),\,\,a\otimes b\mapsto\sum b_{(-1)}\otimes(a\otimes b_{0})$ and $\Upsilon:A\otimes B\longrightarrow(A\otimes B)\otimes Q,\,\,a\otimes b\mapsto\sum(a\otimes b_{0})\otimes b_{(1)}$. Moreover, one checks that $A\otimes B\in{}^{Q}_{Q}{\cal LA}^{Q}_{Q}$, hence we have an L-R-twisting datum for $A\otimes B$. The corresponding L-R-twisted product is:

[TABLE]

for all $a,a^{\prime}\in A$ and $b,b^{\prime}\in B$, and this is exactly the multiplication of the generalized right twisted smash product. and this is exactly the multiplication of the L-R-smash product $A\diamondsuit^{Q}_{l}B$.

Theorem 4.6. With notation as above, let $Q$ be a regular multiplier Hopf algebra with bijective antipode $S$. Let $A\in{}^{Q}_{Q}{\cal LA}^{Q}_{Q}$. Then $A$ is a left $Q\otimes Q^{op}$-Long module algebra (see Example 4.2). Moreover, the corresponding twisted algebras $(A,\bullet)$ and $(A,\circledast)$ are isomorphic, and the isomorphism is defined by:

[TABLE]

In particular, we obtain $A\diamondsuit^{Q}_{r}A\cong A\star^{Q}_{r}A$ and $A\diamondsuit^{Q}_{l}A\cong A\star^{Q}_{l}A$.

Proof. We only prove that $\alpha$ is an algebra isomorphism. It is easy to check that $\alpha\alpha^{-1}=\alpha^{-1}\alpha=id$. Hence we only have to check that $\alpha$ is multiplicative:

[TABLE]

and

[TABLE]

and the proof is finished. $\blacksquare$

ACKNOWLEDGEMENT

The author would like to thank Professor A. Van Daele for his helpful comments on this paper. The work was partially supported by the NSF of China (No. 11371088) and the NSF of China (No.11571173).

REFERENCES

[De] L. Delvaux. Twisted tensor product of multiplier Hopf (*-)algebras. J. of Algebra 269 (2003): 285-316.

[De-VD-W] L. Delvaux, A. Van Daele & S.H. Wang. Bicrossproducts of multiplier Hopf algebras. J. of Algebra 343 (2011): 11-36.

[De-VD] Delvaux L., Van Daele A. The Drinfeld double of multiplier Hopf algebras. J. of Algebra 272(1)(2004): 273-291.

[Dr-VD] Drabant B., Van Daele A. Pairing and quantum double of multiplier Hopf algebras. Algebra Represent. Theory 4(2001): 109-132.

[Dr-VD-Z] B. Drabant, A. Van Daele & Y. Zhang. Actions of multiplier Hopf algebra. Comm. Algebra 27 (1999): 4117-4172.

[P-O] F. Panaite, F. Van Oystaeyen. L-R-smash product for (quasi-)Hopf algebras. J. of Algebra 309 (2007): 168-191.

[VD1] A. Van Daele. Multiplier Hopf algebras. Trans. Am. Math. Soc. 342(2) (1994): 917-932.

[VD2] A. Van Daele. An algebraic framework for group duality. Adv. in Math. 140 (1998): 323-366.

[VD3] A. Van Daele. Tools for working with multiplier Hopf algebras. Arabian Journal for Science and Engineering 33 2C (2008): 505-527.

[VD-VK] A. Van Daele & S. Van Keer. The Yang-Baxter and Pentagon equation. Comp. Math. 91 (1994): 201-221.

[W] S. H. Wang. Doi-Koppinen Hopf bimodules are modules. Comm. in Algebra 29(10)(2001): 4671-4682.

[W-L] S. H. Wang & J. Q. Li. On twisted smash products for bimodule algebras and the Drinfeld double. Comm. in Algebra 26(8)(1998): 1435-1444.