1. Introduction
Hopf algebras originated from the study of the homology of Lie groups and have a natural relationship with groups. Hopf algebras could be seen as an important generalization of groups. In fact, group algebra (a vector space with the basis being a group) is a class of significant Hopf algebra and plays a key role in the theory of Hopf algebra. Quasi-bialgebras and quasi-Hopf algebras have been introduced by Drinfeld in [
1], in connection with the Knizhnik-Zamolodchikov system of partial differential equations, and have been used in several branches of mathematics and physics. In a quasi-bialgebra
H, the comultiplication is not coassociative but is quasi-coassociative in the sense that the comultiplication is coassociative up to conjugation by an invertible element
. Equivalently the representation category of
H is not a strict monoidal category while the reassociation is not trivial. If we draw our attention to the category of co-representations of a coalgebra with non-associative multiplication, we get the concepts of dual quasi-bialgebra and dual quasi-Hopf algebra. These notions have been introduced by Majid in [
2] to prove a Tannaka-Krein type theorem for quasi-Hopf algebras.
For a dual quasi-Hopf algebra H, the category of right H-comodules is monoidal with the usual tensor product. The difference between a dual quasi-Hopf algebra and a Hopf algebra lies in the fact that the associativity of tensor product in the category is not trivial but modified by an invertible element . Consequently, the multiplication of H is no longer associative.
In [
3], the left Yetter-Drinfeld module over quasi-Hopf algebras was first constructed by S. Majid with the help of the isomorphism between the category of Yetter-Drinfeld modules and the center of the representation category. Subsequently, Bulacu, Caenepeel, and Panaite in [
4] introduced all kinds of Yetter-Drinfeld modules and showed that the category of finite dimensional Yetter-Drinfeld modules is rigid. Following the ideas of S. Majid, in [
5] Balan introduced the notion of left-left Yetter-Drinfeld modules over dual quasi-Hopf algebra and studied its Galois extension. Later on, Ardizzoni in [
6] introduced another form of Yetter-Drinfeld module through the isomorphism between the category of Yetter-Drinfeld modules and the category of Hopf bimodules, and characterized as bosonizations the dual quasi-bialgebras with a projection onto a dual quasi-bialgebra. In [
7], the authors introduced the left-left and right-left Yetter-Drinfeld modules and constructed a quantum cocommutative coalgebra in the category of Yetter-Drinfeld modules. In [
8], the authors studied coquasitriangular pointed dual quasi-Hopf algebras and braided pointed tensor categories via the quiver approaches, and classified the Hopf quivers whose path coalgebras admit coquasitriangular dual quasi-Hopf algebras.
Motivated by these ideas, a natural question arises: what are the braided monoidal structures of the categories of left-left, left-right, and right-left Yetter-Drinfeld modules, and via the braidings of other properties of these categories, what could we obtain? In this paper, we will continue the study of the category of Yetter-Drinfeld modules over dual quasi-Hopf algebra H. Firstly, we will give the definition of left-right Yetter-Drinfeld modules and describe their braided monoidal structures explicitly. Moreover, we will show that the categories and are isomorphic, even in the situation where H is not finite-dimensional. Then we will show that the category of finite-dimensional left-left Yetter-Drinfeld modules is rigid, and we compute explicitly the canonical isomorphisms in . Finally, as an application, we rewrite the isomorphisms in the case of coquasitriangular dual quasi-Hopf algebra.
This paper is organized as follows. In
Section 2, we will review the basic results of dual quasi-Hopf algebras and monoidal categories. In
Section 3, we will describe explicitly the braided monoidal structures of categories of all the three kinds of Yetter-Drinfeld modules over dual quasi-Hopf algebra
H. In
Section 4, we will show that the category
of finite dimensional left-left Yetter-Drinfeld modules is rigid and give the explicit forms of the left and right duals of any object. In any rigid braided monoidal category, there exist canonical isomorphisms
and
for any object
M. In
Section 5, we will pay attention to the computations of these isomorphisms in
. In
Section 6, as an application, we will recover the isomorphisms in the case of coquasitriangular dual quasi-Hopf algebra.
2. Preliminary
Throughout this article, let k be a fixed field. All algebras, coalgebras, linear spaces, etc. will be over k; unadorned ⊗ means .
2.1. Dual Quasi-Hopf Algebra
Recall from [
9,
10,
11] that a dual quasi-bialgebra
H is a coassociative coalgebra with comultiplication
and counit
together with coalgebra morphisms
(the multiplication, we write
) and
(the unit, we write
), and an invertible element
(the reassociation), such that for all
the following relations hold
H is called a dual quasi-Hopf algebra if, moreover, there exists an anti-morphism
s of the coalgebra
H and elements
such that for all
,
Throughout this paper, we will always assume that
s is a bijective. It follows from the axioms that
and
. Moreover (
3) and (
4) imply that
Together with a dual quasi-Hopf algebra , we also have and as dual quasi-Hopf algebras. The dual quasi-Hopf structures are obtained by putting , and . , , , , , , . Here .
We recall that an invertible element
satisfying
, induces a twisting transformation
For a Hopf algebra, one knows that the antipode is an anti-algebra morphism, i.e.,
. For a dual quasi-Hopf algebra, this is true only up to a twist, namely, there exists a twist transformation
such that for all
,
where
g denotes the convolution inverse of
f.
The element
f can be computed explicitly. For all
, set
Define elements
by
Then
f and
g are given by the following formulae:
The elements
and the twist
f fulfill the relations
The corresponding reassociation is given by
2.2. Coquasitriangular Dual Quasi-Hopf Algebra
Recall from [
12] that a coquasitriangular dual quasi-Hopf algebra is a dual quasi-Hopf algebra
H with an invertible element
satisfying
for all
.
Let
be a coquasitriangular dual quasi-Hopf algebra. Define
Hom
by
for all
. It is proved in [
12] that
u is invertible with the inverse given by
Moreover
u satisfies the following identities (see [
12]):
2.3. Monoidal Categories and Center Construction
A monoidal category means a category
with objects
etc., a functor
equipped with a natural transformation consisting of functorial isomorphism
satisfying a pentagon identity, and a compatible unit object
I and associated functorial isomorphisms (the left and the right unit constraints,
and
, respectively.) Now if
and
are monoidal categories then, roughly speaking, we say that
is a monoidal functor if it respects the tensor products (in the sense that for any two objects
there exists a functorial isomorphism
such that
respects the associativity constraints), the unit object and the left and right unit constraints (for a complete Definition see [
2]).
If
H is a dual quasi-Hopf algebra, then the categories
and
are monoidal categories. The associative constraint on
is the following: for any
, and
,
is given by
On
, the associative constraint is given by
Let
be a monoidal category, and
.
is called the left dual of
V, if there exist two morphisms
and
such that
is called a right dual of
V if there exist two morphisms
and
such that
is called a rigid monoidal category if every object of
has a left and right dual. The category
of finite dimensional modules over a dual quasi-Hopf algebra
H is rigid. For
,
Hom
with left coaction
. The evaluation and co-evaluation are given by
where
is a basis in
V with dual basis
.
The right dual
of
V is the same dual vector space equipped with the left
H-comodule structure given by
and
For a braided monoidal category , let be equal to as a monoidal category, with the mirror-reversed braiding .
Following [
3], the left weak center
is the category with the objects
, where
and
is a family of natural transformations such that
and for all
,
A morphism between
and
consists of
in
such that
is a prebraided monoidal category. The tensor product is
with
and the unit is
. The braiding
s on
is given by
The center is the full subcategory of consisting of objects with a natural isomorphism. is a braided monoidal category.
The right weak center
is the category with the objects
, where
and
is a family of natural transforms such that
and
for all
. A morphism between
and
consists of
in
such that
is a prebraided monoidal category. The unit is
and the tensor product is
with
The braiding
d is given by
The center
is the full subcategory of
consisting of objects
with
a natural isomorphism.
is a braided monoidal category.
Let
be a monoidal category. Then we have a second monoidal structure on
, defined as
where
and
given by
.
If c is a braiding on , then , defined by is a braiding on .
It is obvious that
Proposition 1. ([
4]).
Let be a monoidal category. Thenas the prebraided monoidal category, andas a braided monoidal category. Definition 1 ([7]). Let H be a dual quasi-bialgebra.
- (1)
A k-space M is called a left-left Yetter-Drinfeld module if M is a left H-comodule (denote the left coaction by , ) and H acts on M from the left (denote the left action by ) such that the following conditions hold: for all and . The category of left-left Yetter-Drinfeld modules over H is denoted by with the morphisms being left H-linear and left H-colinear.
- (2)
A right-left Yetter-Drinfeld module is a left H-comodule M together with a right H-action · on M such that for all , The category of right-left Yetter-Drinfeld modules over H is denoted by with the morphisms being right H-linear and left H-colinear.
3. Yetter-Drinfeld Modules over a Dual Quasi-Hopf Algebra
In this section, we will describe braided monoidal structures of the categories of Yetter-Drinfeld modules over dual quasi-Hopf algebra H and show that the categories and are isomorphic.
Let
H be a dual quasi-Hopf algebra. Recall from [
5], for all
, define elements
in
by
Lemma 1. Let H be a dual quasi-Hopf algebra. For all ,andMoreover we have the following formulae Proof. The identities (
24)–(
26) come from [
7]. Since
is also a dual quasi-Hopf algebra, by (
26) we could obtain (
27). □
Proposition 2. Let H be a dual quasi-Hopf algebra, , and a k-linear map satisfying (19) and (20). Then (21) is equivalent tofor all . Proof. The proof is similar to that of [
5]. □
Example 1 Let be a coquasitriangular dual quasi-Hopf algebra. Then any left H-comodule M is a left Yetter-Drinfeld module over H. Indeed for all , defineThen for the relation (19)And for the relation (21) Proposition 3. ([
5]).
Let H be a dual quasi-bialgebra and the category of left H-comodules. Then we have category isomorphism . The action of
H on the tensor product
of two left-left Yetter-Drinfeld modules
M and
N is given by
for all
. The braiding is given by
Furthermore we have the following result.
Theorem 1. Let H be a dual quasi-Hopf algebra. The braiding c is invertible with the inverse given by Proof. For all
,
That is,
. Similarly
. The proof is completed. □
We also introduce left-right Yetter-Drinfeld modules in the following definition.
Definition 2. Let H be a dual quasi-Hopf algebra. A left-right Yetter-Drinfeld module is a right H-comodule M together with a left H-action · on M such that for all ,The category of left-right Yetter-Drinfeld modules over H is denoted by with the morphisms being left H-linear and right H-colinear. Theorem 2. Let H be a dual quasi-bialgebra. Then we have the following category isomorphisms:If H is a dual quasi-Hopf algebra, then these three weak centers are equal to the centers. Proof. The proof is straightforward and left to the reader. □
The prebraided monoidal structure on
induces a monoidal structure on
. We find that the action on
of two right-left Yetter-Drinfeld modules
M and
N are given by
for all
, and
.
The braiding
is given by
In the case when
H is a dual quasi-Hopf algebra, the inverse of
is given by
The prebraided monoidal structure on
: for
, the action on
is given by
and
the braiding is the following:
In the case when
H is a dual quasi-Hopf algebra, the inverse of
is given by
Proposition 4. We have an isomorphism of monoidal categorieswhere F acts on objects and morphisms as identity, and the right H-coaction is given by . Similarly, we have Proof. We only need to verify that
F preserves the monoidal structure. For all objects
,
The associativity constraint
is defined as
As for the monoidal structure on
, we have
. Then
The associativity constraint
is defined as
The proof is completed. □
Proposition 5. Let H be a dual quasi-Hopf algebra. Then we have the following isomorphisms of braided monoidal categories: Proof . By Proposition 1 and Proposition 4, we obtain
and
□
Proposition 6. ([
4]).
Let be a monoidal category. Then we have a braided isomorphism of braided monoidal categories , given by Of course, the conclusion holds for the right center. By this isomorphism, we have the following result.
Proposition 7. Let H be a dual quasi-Hopf algebra, and the category with the braidingThen we have an isomorphism of braided monoidal categoriesdefined as follows. For , as a left H-comodule; the left H-action is given byfor all , where · is the right action of H on M. The functor T sends a morphism to itself. Proof. The functor
T is just the composition of the isomorphisms
For
, we compute the corresponding left-left Yetter-Drinfeld module structure on
M is the following:
as claimed. □
In the same way, we have the following result.
Proposition 8. Let H be a dual quasi-Hopf algebra. Then the categories and are isomorphic as braided monoidal categories.
4. The Rigid Braided Category
It is well known that the category of finite dimensional Yetter-Drinfeld modules over a Hopf algebra with a bijective antipode is rigid. Since is rigid, the same result holds for the category of finite dimensional Yetter-Drinfeld modules over a dual quasi-Hopf algebra. In this section, we will give the explicit forms.
Proposition 9. ([
4]).
Let be a rigid monoidal category. Then the weak left (respectively right) center of is a rigid braided monoidal category. For Example, for any object
,
, with
given by the following composition:
Lemma 2. Let H be a dual quasi-Hopf algebra. Then for all , the following relations hold: Proof. By the Definition of
, it is easy to verify (
30). Then for all
,
as needed. The proof is completed. □
Theorem 3. Let H be a dual quasi-Hopf algebra. Then is a braided monoidal rigid category. For a finite-dimensional left-left Yetter-Drinfeld module M with basis and dual basis , the left and right duals and are equal to Hom as a vector space, with the following H-action and H-coaction:
for all .
Proof. The left
H-coaction on
viewed as an object in
is the same as the left
H-coaction on
viewed as an object in
. Now we compute the left
H-action using (
29). For all
,
as claimed. The structure on
can be computed similarly. The proof is completed. □
5. The Canonical Isomorphisms in
If
is a rigid braided monoidal category, then for any objects
, there exist two canonical isomorphisms in
In this section, we aim to give the explicit forms of the above isomorphisms in the particular case
Let
be a rigid monoidal category and objects
and
is a morphism in
. Following [
13] we can define the transposes of
as the compositions:
From [
4] we have two isomorphisms
and
. Both isomorphisms are natural in
M.
and its inverse could be described explicitly as follows:
We also have a natural isomorphism
, which can be described as follows, see [
14] for details.
Thus the functors
and
are naturally isomorphic, and we conclude that
Now we will apply these results to the particular case when
(1) For all
,
where
is a basis of
M and
its dual basis. By a similar computation, we have
(2) For
we have
where
is a basis of
M with dual basis
in
, and
are bases of
, and
is the image of
under the canonical map
. Moreover the morphism
is defined by the same formula as
. The maps
is given by
Moreover the morphisms
and
are defined the same as
and
, respectively. (2) As to
, for all
, we have
By a similar computation, the inverse map
is given by
where
. Thus we obtain the following result.
Proposition 10. Let H be a dual quasi-Hopf algebra and . Then is an isomorphism of Yetter-Drinfeld modules. Explicitly, is given byfor all . The inverse of is given by , that is,Similarly is an isomorphism of Yetter-Drinfeld modules. Explicitly we havefor all . The inverse of is given by for all , where .
Proof. Similarly, we could obtain and and the details are left to the reader. The proof is completed. □
Let
be a rigid monoidal category. For any objects
, there exists two isomorphisms
where
is the composition
with the inverse
given by the composition
Moreover if
is braided, then we have the following isomorphism
Before proceeding we need the following Lemma.
Lemma 3. Let H be a dual quasi-Hopf algebra. The following relations hold: Proof. We only prove (
36), (
38) and (
40), and the rest can be proved similarly.
(1) By the relations (
8) and (
14), for all
,
Hence
Using the relation (
12), we have
(2) Similarly we could obtain the relation (
38).
(3) For all
,
That is
The proof is completed. □
Proposition 11. Let H be a dual quasi-Hopf algebra and . Denote and the dual bases in M and , and and dual bases in N and . Define the map byThen is an isomorphism in . The inverse of is given byIn a similar way, the isomorphism is given bywith its inverse Proof. For all
and
,
Similarly we obtain
for all
. Denote
, then
For all
and
, we compute
Obviously the inverse of
is
. For all
,
, we compute
By similar computations, we could obtain the identities (
44) and (
45). The proof is completed. □
6. Application
Let be a coquasitriangular dual quasi-Hopf algebra. Just as shown in Example 1, any left H-comodule is a left Yetter-Drinfeld module. In this section, we will rewrite the canonical isomorphisms.
As an immediate consequence of Proposition 10, we have
Proposition 12. Let be a coquasitriangular dual quasi-Hopf algebra and M a finite-dimensional left H-comodule. Then and as left H-comodules.
Proof. We have seen that
M is an object in
, so
and
as left Yetter-Drinfeld modules. Thus
and
as left
H-comodules. By a direct computation, we have that
is given by
with the inverse
After similar computations, we obtain that
is given by
with its inverse
□
Proposition 13. Let be a coquasitriangular dual quasi-Hopf algebra and two finite-dimensional left H-comodules. Then and as left H-comodules.
Proof. The result is a direct consequence of Proposition 11. Now we will give these isomorphisms explicitly.
and
Similarly we have
and
The proof is completed. □
7. Conclusions
Yang-Baxter equation (or star-triangle relation) is a consistency equation that was first introduced in the field of statistical mechanics, and it takes its name from the independent work of C. N. Yang from 1968, and R. J. Baxter from 1971. In mathematical physics, one of the most classic problems is to find the solutions to the Yang-Baxter equation. Braided monoidal categories have been playing an essential role since they could supply such solutions. Hence the attention of mathematicians was naturally drawn to the construction of braided monoidal categories. V. Drinfeld developed an elegant theory that the category of Yetter-Drifeld modules over any Hopf algebra turns out to be a braided monoidal category, thus supplying solutions to the Yang-Baxter equation. Since then, the idea was extended to a more general Hopf algebra structure. In this paper, we mainly focus on the properties of the Yetter-Drifeld category over dual quasi-Hopf algebras. Concretely, we firstly describe explicitly the braided monoidal structures of three kinds of Yetter-Drifeld categories; then prove that the subcategory of finite dimensional Yetter-modules is rigid, and for any object, M, give the Yetter-Drinfeld module structures on and ; finally, compute the canonical isomorphisms in , and present an application in coquasitriangular dual quasi-Hopf algebras case.
The results obtained in our paper indeed enrich the theory of the Yetter-Drinfeld category and could lay the foundation for further research on dual quasi-Hopf algebras, for example, the constructions of the category of Yetter-Drinfeld-Long bimodules and Drinfeld double of dual quasi-Hopf algebra. Moreover, The results could also be applied to the research on the theory of category, especially to the monoidal category, braided category, fusion category, and even to the construction of more complicated crossed group category.
Author Contributions
Conceptualization, Y.N. and D.L.; methodology, D.L.; software, Y.N.; validation, Y.N. and D.L.; formal analysis, Y.N.; investigation, D.L.; resources, D.L.; data curation, X.Z.; writing—original draft preparation, D.L.; writing—review and editing, Y.N.; visualization, Y.N.; supervision, D.L.; project administration, D.L. and X.Z.; funding acquisition, D.L. and X.Z. All authors have read and agreed to the published version of the manuscript.
Funding
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11901240, 12001174).
Data Availability Statement
Not applicable.
Acknowledgments
The authors would like to express their gratitude to the anonymous referees for their very helpful suggestions and comments, which lead to the improvement of our original manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
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