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

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].
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Advanced Operator Algebra Research
Twisted Algebras of Multiplier Hopf (∗-)algebras
**Shuanhong Wang
**Department of Mathematics, Southeast University
Nanjing, Jiangsu 210096, P. R. of China
E-mail: [email protected]
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 be an associative algebra with a nondegenerate product. If 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 be a finite set of elements in . Then there exist elements so that for .
A multiplier of the algebra is a pair of linear mappings in such that for all . The set of multipliers of is denoted by . It is a unital algebra which contains as essential ideal through the embedding . Therefore and for all and . Hence we will frequently use the identification and . If is unital then . If is a -algebra then is a -algebra through where for any , . Since the multiplication of is supposed to be nondegenerate a multiplier of is uniquely determined by its first or second component. For a tensor product of two algebras and one obtains the canonical algebra embeddings .
Let be an algebra with a non-degenerate product. An algebra homomorphism is called a comultiplication on if
(i) and for all (some times we call this condition a ”coving theory”);
(ii) on .
We call with a comultiplication a multiplier Hopf algebra if the linear maps on are bijective. Moreover, we call regular if , where is the flip, is again a comultiplication such that is also a multiplier Hopf algebra. If is a -algebra, we call 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 when . The problem is that is not in in general. By covering theory, we know however that for all . We can write (cf. [Dr-VD-Z, VD3]) for this. Now, we know that there is an element such that and we can think of to stand for . Of course, this is still dependent on . But we know that for several elements , we can use the same .
Let be a regular multiplier Hopf algebra (i.e. with a bijective antipode). By a left -module , we always mean a unital module (sometimes, we also say that is a left action of on ). This means that . For all , we have an element such that . Observe that for a unital algebra , this condition is automatically satisfied. Similarly, a right -module can be defined.
Let denote a regular multiplier Hopf algebra and an associative algebra with or without identity. Assume that is a left action of on making into a left -module algebra, i.e., we have for all and . In this formula, is covered by (through the action) and is covered by (through the action). Similarly, assume that is a right action of on making into a right -module algebra, i.e., one has , for all and . Then is called a -bimodule algebra, if it is a unital left -module and a unital right -module such that , for all and .
Suppose that is a left coaction of on making a left -comodule algebra. More precisely, is an injective homomorphism so that for all and , we have and are in . We use the Sweedler notation for these expressions, e.g. . We say that the multiplier is covered by . Now the further requirement makes sense: for all . In fact, the expression makes sense when it is understood that is replaced by . We have more or less the same rules for covering. We need to cover the factor and possibly also (and so on) by elements in , left or right. In this case however, one cannot cover the factor . Similarly, the right coaction of on can be defined. Then is called a --bicomudule algebra if
[TABLE]
for all and .
Let be multiplier Hopf algebras. Denote the category of all left -module algebras by , denote by the category of all right -module algebras , and denote by the category of all --bimodule algebras. Similarly, denote the category of all left -comodule algebras by , denote by the category of all right -module algebras , and denote by the category of all --bicomodule algebras.
Let and be a dual pair of algebraic quantum groups over . By Pontryagin duality, identifying , we have a natural dual pairing is written as
[TABLE]
Note that the antipode of and is given by and the antipode of by . Also, , and .
Obviously, after a permutation of tensor factors a left coaction of may always be viewed as a right coaction by and vice versa.
Next, we recall that there is a one-to-one correspondence between right (left) coactions of on and left (right) actions, respectively, of on given for and by
[TABLE]
and
[TABLE]
As a particular example we recall the case with . In this case we denote the associated left and right actions of on by and , 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 be algebras and suppose that there are two -linear maps and such that for
[TABLE]
Here we write and for all and , and , , , works on the first and the third componets as and for any -vector space .
One can define a generalized L-R-twisted tensor product as spaces with a new multiplication given by the formula:
[TABLE]
for all and . Here, works on the first and the forth componets as .
Proposition 2.1.1. The above multiplication on is associative.
Proof. For all and . We compute
[TABLE]
and
[TABLE]
This completes the proof.
Remark 2.1.2. When is an identity map, then a generalized L-R-twisted tensor product becomes a twisted tensor product . As a space, with the following product defined by the formula:
[TABLE]
for all and . Here, .
If and and have a unit , we suppose that satisfies and for all and . Then is the unit of . Furthermore, the maps and are algebra embeddings. For more details on the above results, we refer to [VD-VK].
Definition 2.1.3. Let and be two algebras without unit, but with non-degenerate products. Let and be two -linear maps. Then one can define the following two -linear maps and from to :
[TABLE]
for all and .
Proposition 2.1.4. Let and be two algebras without unit, but with non-degenerate products. If and are bijective such that (1): implies that and (2): implies that for all and , then the product in is non-degenerate.
Proof. Suppose that there is an element such that for all . 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 .
Similarly, by Eq.(2.1), (2.4), (2.5) and the assumption (2), one can show that for all implies that .
Corollary 2.1.5. Let and be two algebras without unit, but with non-degenerate products.
(1) ([De, Proposition 1.1]) If is bijective, then the product in is non-degenerate.
(2) If is bijective, then the product in is non-degenerate.
Proof. (1) In this case, . Thus the conditions (1): implies that and (2): implies that for all and , become that (1): implies that and (2): implies that for all and . We now explain how these two conditions hold. In fact, for the condition (1), since the product on is non-degenerate, implies that and thus since the product on is non-degenerate. Similarly, the condition (2) holds too.
(2) In this case, . The check is similar to the one in the part (1).
Theorem 2.1.6. Let and be Hopf algebras with two -linear maps and such that Eq.(2.1)-(2.5) hold and
[TABLE]
Here we write and for all and .
Then is a bialgebra, where the comultiplication, and are given as
[TABLE]
for all and .
Furthermore, if
[TABLE]
then is a Hopf algebra with given by
[TABLE]
Proof. From Proposition 2.1.1, we know that is an algebra with unity . Clearly is coassociative and is counity.
Take and . We now prove that is a homomorphism.
[TABLE]
From the counitary property (2.8) we obtain that is a homomorphism on . Therefore, is a bialgebra.
We show that .
[TABLE]
Similarly, by Eq.(2.1), (2.4), (2.5) and Eq.(2.12), we can get
[TABLE]
Thus, is a Hopf algebra.
Remark 2.1.7. In the . We have
(1) when is the identity map, i.e, , 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 is the flip map, i.e., , then the conditions (2.11) and (2.12) hold too.
(3) When and are not trivial, we will give the example (see Example 2.3.1 (3)).
Proposition 2.1.8. Let and be Hopf -algebras with two -linear maps and such that and satisfy all the conditions of Theorem 2.0.6. If furthermore
[TABLE]
then is a Hopf -algebra
Proof. By Theorem 2.1.6, one has that is a Hopf algebra. By the conditions that and are involutions on , we have that is a -algebra when . In what follows, we check that is a -homomorphism on .
[TABLE]
This finishes the proof.
Proposition 2.1.9. Let , and be multipliers. Then
(1) we have that where is defined by
[TABLE]
for all and .
(2) One has that where is defined as follows:
[TABLE]
for all and , where we write and .
Proof. (1) For all and .
[TABLE]
(2) Similarly.
This finishes the proof.
Definition 2.1.10. Let and be multiplier Hopf algebras with two -linear maps and as described above. One defines:
[TABLE]
and
[TABLE]
We now prove that these candidates and defined by and , respectively can be used to define a good comultiplication.
Proposition 2.1.11. Take the notations as above. For all and , define
[TABLE]
and
[TABLE]
Then is a two-sided multiplier of , where . Furthermore, is coassociative on .
Proof. For all and , we compute.
[TABLE]
Similarly, by Eq.(2.2), (2.3) and (2.5) we have
[TABLE]
Firstly, for , we prove that is a two-sided multiplier of . In fact, notice that and recall from Proposition 2.1.8 that for and , we can form the multiplier . We can easily check that . It is clear that in the case that and are Hopf algebras, .
Then in order to check that is coassociative on , we have to prove that
[TABLE]
This calculations are straightforward by making use of the expressions
[TABLE]
and
[TABLE]
and the coassociativity of and .
This completes the proof.
We now have the main result of this section as follows.
Theorem 2.1.12. Let and be multiplier Hopf algebras with two -linear maps and such that Eq.(2.1)-(2.5) hold. If and are bijective such that (1): implies that and (2): implies that for all and , and
[TABLE]
then is a multiplier Hopf algebra, where the multiplication, , and are given as
[TABLE]
for all and .
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 and we have that is an associative algebra with non-degenerate product.
(2) From Proposition 2.1.10 we conclude that is coassociative on .
(3) We prove that is a homomorphism. For all and , we have
[TABLE]
(4) Define the counit on by . We have to prove that for all and
(i) ;
(ii) .
We prove (i), the proof of (ii) is similar. From Proposition 2.1.10 we have that for and
[TABLE]
where and .
So,
[TABLE]
(5) Because is surjective and is a homomorphism, is a homomorphism. This can be proved in a similar way as in [VD1, Lemma 3.5].
(6) On , define the antipode which is an invertible map. We have to prove that for all and
(i) ;
(ii) .
We prove (i), the proof of (ii) is similar. In fact, start again from the following equation:
[TABLE]
where and .
Then we have
[TABLE]
(7) Because is surjective and is a homomorphism, 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 is a (regular) multiplier Hopf algebra.
Proposition 2.1.13. Let and be multiplier Hopf -algebras with two -linear maps and such that and satisfy all the conditions of Theorem 2.1.12. If furthermore
[TABLE]
then is a multiplier Hopf -algebra.
Proof. Straightforward.
Proposition 2.1.14. Let and be multiplier Hopf algebras as in Theorem 2.1.12. Let (resp. ) be a right integral on (resp. ). Then is a right integral on the multiplier Hopf algebra .
Proof. For all and ,
[TABLE]
This finishes the proof.
Proposition 2.1.15. Let and be multiplier Hopf algebras as in Theorem 2.1.12. Let (resp. ) be a left integral on (resp. ) with associated modular element (resp. ). Then the multiplier is the modular element in associated to .
Proof. Recall from [VD2] that the modular element in for a multiplier Hopf algebra is given by when . Now, for the multiplier Hopf algebra , we have that modular element associated to is given as
[TABLE]
We claim that
For all and , we have
[TABLE]
This completes the proof.
2.2 Generalized smash products
Let be a regular multiplier Hopf algebra and let be a right -module algebra and a right -comodule algebra. Then there are a -linear maps for and . By Remark 2.1.2, we may define a right smash product as -vector space with multiplication
[TABLE]
for , where we use the notation in place of to emphasize the new algebraic structure.
Let be a regular multiplier Hopf algebra and let be a left -module algebra and a left -comodule algebra. Then there are another -linear maps for and . Similarly, by Remark 2.1.2, denotes the associative algebra structure on given by
[TABLE]
for , containing and as subalgebras.
We have obvious embeddings of and in the multiplier algebra of . And we note that with for . Similar statements hold in .
Example 2.2.1 (1) Given a left coaction of on with dual right -action . One has the right smash product to be the vector space with associative algebra structure given for and by
[TABLE]
(2) Similarly if is a right coaction of on with dual left -action , then the multiplication of the algebra is given by
[TABLE]
for and
We remark that any right smash product can be identified with an associated left smash product , where is the right coaction given by
[TABLE]
with being the permutation of tensor factors. In fact we have
Lemma 2.2.2. Let and be a pair of left and right coactions, respectively, related by (2.2.3). Then we have
[TABLE]
Proof. Let and be the left and right actions dual to and , respectively. The equation (2.2.3) is equivalent to for all and .
Define a linear map as
[TABLE]
It is easy to check that its inverse is given by
[TABLE]
for and .
The remaining thing is straightforward.
Let be Hopf algebras and let be a right -module algebra and a right -comodule algebra. Let be a bialgebra map. Then there are a -linear maps for and . By Remark 2.1.2, we can define a right smash product as -vector space with multiplication
[TABLE]
for , where we use the notation in place of to emphasize the new algebraic structure.
Then the proof of the following proposition is straightforward.
Proposition 2.2.3. With the above notation. is an associative algebra.
Similarly, let be a left -module algebra and a left -comodule algebra. Let be a bialgebra map. Then there are a -linear maps for and . By Remark 2.1.2, denotes the associative algebra structure on given by
[TABLE]
for , containing and as subalgebras.
2.3 Generalized twisted smash products
Let be a regular multiplier Hopf algebra. Let be a -bimodule algebra, i.e, and let be a -bicomodule algebra, i.e., . In this section, we consider an algebra that is a generalized smash product of and . 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 as -vector space with multiplication
[TABLE]
for .
Similarly denotes the associative algebra structure on given by
[TABLE]
for .
Proposition 2.3.1 and as above are associative algebras.
Proof. For , the twist map is given by the formula: for and . By Remark 2.1.2, is an associative algebra.
Similarly for and the proof is finished.
The algebra (resp. ) is called a generalized right (resp. left) smash product, which is denoted by (resp. ). Just as in the case of smash products, we have obvious embeddings of and in the multiplier algebra of (resp. ) and if we identify these two algebras with their images in (resp. ), we see that (resp. ) is the linear span of elements (resp. ) with and and that we have the commutation rules
i) and commute,
ii)
(resp. ), for .
Therefore we can view (resp. )as the algebra generated by and 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 is trivial, then we obtain for simply the tensor product algebra . If the left action and coaction of on and are trivial, respectively, we obtain the right smash product .
(2) Let denote an algebraic quantum group. Then is the -bimodule algebra. Let be a -bicomodule algebra. Then we have two generalized twisted smash products and .
(3) Let denote a regular multiplier Hopf algebra with antipode and a -bimodule algebra. Then the generalized twisted smash product is isomorphic to an ordinary smash product , where the left -action on is now given by , for all and . 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 be a multiplier Hopf algebra, a -bimodule algebra, i.e., and a -bicomodule algebra, i.e., . Define and as and , respectively, for all and . Then, by Eq. (2.1.1), we have the generalized L-R-twisted tensor product with the multiplication by
[TABLE]
for and .
Similarly, Define and for all and . Then, by Eq. (2.1.1), we can define as -vector space with multiplication
[TABLE]
for .
Example 2.4.1. (1) Let be a right -module algebra. Then becomes an -bimodule algebra, with left -action given via . In this case the multiplication of becomes
[TABLE]
for , hence in this case coincides with the generalized smash product .
(2) We note that itself is a -bicomodule algebra. So, in this case, the multiplication of specializes to
[TABLE]
for . If the left -action is trivial, then coincides with the smash product
(3) If we consider a usual Hopf algebra , and assume that is a -bimodule algebra. Define and for all and . Then, by Eq. (2.1.1), we can define as -vector space with multiplication
[TABLE]
for . If and , 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. and as above are associative algebras.
The proof of this result is straightforward.
Proposition 2.4.3. Let be a -bimodule algebra and let be a -bicomodule algebra. Then and .
Proof. Define as
[TABLE]
for all and . First, it is not hard to check that the map has the inverse given by
[TABLE]
Then we check that is a homomorphism as follows.
[TABLE]
for .
Similarly for . This completes the proof.
Example 2.4.4. (1) Let be a regular multiplier Hopf algebra and let be a -bicomodule algebra. If is a left -module algebra regarded as an -bimodule algebra with trivial right -action, then and both coincide with , and the isomorphism is just the identity.
(2) If , the maps and become:
[TABLE]
for all and .
Let now be a multiplier Hopf algebra, a -bimodule algebra and a -bicomodule algebra. Let be an algebra in the Yetter-Drinfeld category (see [De]) that is is both a left -module algebra and a left -comodule algebra. These two structures are crossed via the Yetter-Drinfeld compatibility condition in the following sense. For all and we require
[TABLE]
Consider first the generalized smash product , an associative algebra. From Eq.(2.4.4), it follows that becomes a -bimodule algebra, with -actions
[TABLE]
for all and , hence we may consider the algebra . Similarly, for .
Meanwhile, consider the generalized smash product , an associative algebra. Using the condition Eq.(2.4.4), one can see that becomes a -bicomodule algebra, with -coactions:
[TABLE]
for all and , hence we may consider the algebra .
Proposition 2.4.5. We have an algebra isomorphism , given by the trivial identification.
Proof. We compute the multiplication in as follows:
[TABLE]
The multiplication in is:
[TABLE]
Hence the two multiplications coincide, completing the proof.
Since the L-R-smash product coincides with the generalized smash product if the right -action is trivial, we also obtain:
Corollary 2.4.6. If are as above and is a left -module algebra, then we have an algebra isomorphism , given by the trivial identification.
Let be a regular multiplier Hopf algebra. Recall that the Drinfel’d double (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 is just the twisted smash product . By transferring the whole structure of via the map , we can thus obtain a new realization of , having the L-R-smash product for the algebra structure.
Let be a multiplier Hopf algebra. From [De-VD], an invertible element is called a Drinfeld twist (or a gauge transformation) if
[TABLE]
If is a Drinfeld twist with inverse , then we can define a new multiplier Hopf algebra (see, [Wa1]) with the same multiplication and counit as , for which the comultiplication and antipode are given by, for
[TABLE]
where is an invertible element of with the inverse .
Remark: (1) It is easy to get that
[TABLE]
and
[TABLE]
(2) Let be an invertible element such that . If is a twist for then so is . The twists and are said to be gauge equivalent.
Let be a multiplier Hopf algebra, a -bimodule algebra and a Drinfeld twist. If we introduce on another multiplication, by for all , and denote this structure by , then is a -bimodule algebra, with the same -actions as . We have
Proposition 2.4.6. Take the notations as above. Then is a -bimodule algebra.
Proof. We first want to show that the product in is associative and non-degenerate. Furthermore, remains the unit in .
We compute
[TABLE]
for all
Then, one has to show that is a left -module algebra.
We note that
[TABLE]
for all
Similarly, one can check that is a right -module algebra and that is a -bimodule. This finishes the proof.
Suppose that we have a left -comodule algebra ; then on the algebra structure of one can introduce a left -comodule algebra structure (denoted by in what follows) putting the same -coaction as . Similarly, if is a right -comodule algebra, one can introduce on the algebra structure of a right -comodule algebra structure (denoted by in what follows) putting the same -coaction as . One may check that if is a -bicomodule algebra, the left and right -comodule algebras respectively actually define the structure of a -bicomodule algebra on , denoted by .
Then the proof of the following result is straightforward.
Proposition 2.4.7. Take the notations as above. If is a -bicomodule algebra, then is a -bicomodule algebra.
Proposition 2.4.8. Let be a -bimodule algebra and a -bicomodule algebra. Then we have two algebra isomorphisms:
[TABLE]
given by the trivial identification.
Proof. We only compute the multiplication in . Similar to .
[TABLE]
for .
which is the multiplication of
This finishes the proof.
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 be algebras and let and be -linear maps. Then we will write and for all and . One defines two-sided twisted tensor product as spaces with a new multiplication defined by the formula:
[TABLE]
or
[TABLE]
for all and , where denotes the usual flip map on .
Let be multiplier Hopf algebras. Let be in , , and let be in . If we define and , respectively by and for all and , then we have a two-sided twisted tensor product with the multiplication given by
[TABLE]
for , here we write for .
Proposition 3.1.1. defined above is an associative algebra.
Note that, given as above, becomes a -bimodule algebra, with -actions
[TABLE]
for all .
Then we have
Proposition 3.1.2. Let be as above. Then we have the following algebra isomorphisms
(1) ;
(2) ;
(3) ;
(4) .
Proof. (1) We do calculations as follows:
[TABLE]
for .
(2) Similar to (1).
(3) follows (1), (2) and Proposition 2.4.3.
(4) We note that has an induced right -comodule coaction by the right coaction of on , i.e., , for all . Hence, . Similarly, .
This finishes the proof.
3.2 Two-sided L-R-smash products
Let be a multiplier Hopf algebra, a right -comodule algebra, a left -comodule algebra and a -bimodule algebra. Define on a multiplication by the formula
[TABLE]
for .
Then this multiplication yields an associative algebra structure, denoted by .
Note that, given as above, becomes an -bicomodule algebra, with the following structure:
[TABLE]
and
[TABLE]
Proposition 3.2.1. If are as above, then we have the following algebra isomorphisms:
(1) ;
(2) ;
(3) ;
(4) .
Proof. (1) We compute:
[TABLE]
(2) and (3) Obvious.
(4) We note that has an induced -bimodule algebra by the one on . This finishes the proof.
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 be a multiplier Hopf algebra.
(1) Let be a left -module algebra and a left -comodule algebra. is called a left-left -Long module algebra if the following condition holds:
[TABLE]
for all . The category of all left -Long module algebras is denoted by .
(2) Let be a left -module algebra and a right -comodule algebra. is called a left-right -Long module algebra if the following condition holds:
[TABLE]
for all . The category of all left-right -Long module algebras is denoted by .
(3) Let be a right -module algebra and a right -comodule algebra. is called a right-right -Long module algebra if the following condition holds:
[TABLE]
for all . The category of all right-right -Long module algebras is denoted by .
(4) Let be a right -module algebra and a left -comodule algebra. is called a right-left -Long module algebra if the following condition holds:
[TABLE]
for all . The category of all right-left -Long module algebras is denoted by .
Let . If we define a new multiplication on , by
[TABLE]
then this multiplication defines a new algebra structure on . The product is called the left twisted product.
Example 4.2. Let be a regular multiplier Hopf algebra with bijective antipode , a -bimodule algebra and a -bicomodule algebra. Then becomes a left -module algebra with the following structure:
[TABLE]
and becomes a left -comodule algebra with the following structure
[TABLE]
It is easy to check that is a left -Long module algebra with the natural structure. The corresponding twisted product on is
[TABLE]
for all and , and this is exactly the multiplication of the generalized right twisted smash product.
Similarly, for . If we define a new multiplication on , by
[TABLE]
then this multiplication defines a new algebra structure on . The product is called the right twisted product.
Let be a regular multiplier Hopf algebra with bijective antipode , let be a -bimodule algebra and a -bicomodule algebra such that , called four sides Long module category. Then we define a new multiplication on by
[TABLE]
and call it the L-R-twisted product.
Then it is easy to check that is a left -Long module algebra with the natural structure (see Example 4.2). The corresponding twisted product on is
[TABLE]
Proposition 4.3. is an associative algebra.
Proof. We do calculations as follows:
[TABLE]
for all .
This finishes the proof.
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 and respectively 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 . With notation as before. Then the L-R-twisted product can be obtained as a left twisting followed by a right twisting and also vice versa.
Proof. First consider the left twisted product algebra ; it is easy to see that , and the corresponding right twisted product becomes:
[TABLE]
for all .
Similarly, one can start with the right twisted product algebra , for which , and the corresponding left twisted product coincides with the L-R-twisted product.
This finishes the proof.
Example 4.5. Let be a regular multiplier Hopf algebra, a -bimodule algebra and a -bicomodule algebra. Take the algebra , which becomes a -bimodule algebra with actions and , for all , and a -bicomodule algebra, with coactions and . Moreover, one checks that , hence we have an L-R-twisting datum for . The corresponding L-R-twisted product is:
[TABLE]
for all and , 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 .
Theorem 4.6. With notation as above, let be a regular multiplier Hopf algebra with bijective antipode . Let . Then is a left -Long module algebra (see Example 4.2). Moreover, the corresponding twisted algebras and are isomorphic, and the isomorphism is defined by:
[TABLE]
In particular, we obtain and .
Proof. We only prove that is an algebra isomorphism. It is easy to check that . Hence we only have to check that is multiplicative:
[TABLE]
and
[TABLE]
and the proof is finished.
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).
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