Abstract

Let be a BiHom-Hopf algebra and be an -module BiHom-algebra. Then, in this paper, we study some properties on the BiHom-smash product . We construct the Maschke-type theorem for the BiHom-smash product and form an associated Morita context .

1. Introduction

The first instance of Hom-type algebras appeared in the physics literature when looking for quantum deformations of some algebras of vector fields in 1990’s, such as Witt and Virasoro algebras ([1, 2]). This kind of algebras obtained by deforming certain Lie algebras no longer satisfied the Jacobi identity, but a modified version of it involving a homomorphism. Such algebra (called Hom-Lie algebra) was given in [3, 4]. The associative counterpart of Hom-Lie algebras has been introduced in [5] (called Hom-associative algebras), and Hom-analogues of other algebraic structures have been introduced afterwards, Hom-coassociative coalgebras, Hom-bialgebras, Hom-pre-Lie algebras etc.

A categorical approach to Hom-type algebras was considered in [6]. A generalization has been given in [7], where a construction of a Hom-category including a group action led to concepts of BiHom-type algebras. Hence, BiHom-associative algebras and BiHom-Lie algebras, involving two linear maps (called structure maps), were introduced. The main tool to obtain examples of Hom-algebras from classical algebras, the so-called Yau twisting, works perfectly fine also in the BiHom-type case. There is a growing literature on Hom and BiHom-type algebras, and let us just mention the very recent papers [8–12].

Let be a Hopf algebra and an -module algebra; then, as well known, we can construct the smash product algebra (see [13] or [14]). Smash products plays an important role in the lifting method for the classification of finite-dimensional pointed Hopf algebras (see [15]). The Hom-forms of the smash product can be found in the following literature. In [16], the Maschke-type theorem for the Hom-smash product is given, and the Morita context is constructed. In [7], the authors defined the BiHom-smash product and gave some examples. Now, it is natural to ask how to prove the Maschke-type theorem and construct the associated Morita context for the BiHom-smash product?

The main aim of this paper is to give a positive answer to the above questions. We use the same strategy as in the Hom case and get an analogue of the Maschke-type theorem and form an associated Morita context between the BiHom-smash product and its BiHom-subalgebra in the setting of BiHom-Hopf algebras.

This paper is organized as follows. In Section 2, we recall some definitions and basic results related to BiHom-algebras, BiHom-coalgebras, BiHom-bialgebras, BiHom-modules, and module BiHom-algebras. In Sections 3 and 4, we study some properties on the BiHom-smash product . If is a finite-dimensional semisimple BiHom-Hopf algebra, then we construct the Maschke-type theorem for the BiHom-smash product . We also prove that forms an associated Morita context, where is a BiHom-subalgebra of -invariants in .

2. Preliminaries

We work over a base field . All algebras, linear spaces, etc. will be over ; unadorned means . We use Sweedler’s notation for terminologies on coalgebras. For a coalgebra , we write comultiplication , for any .

Definition 1 ([7]). A BiHom-associative algebra is a 4-tuple , where is a linear space and and are linear maps such that , , and for all . The maps and (in this order) are called the structure maps of , and condition (1) is called the BiHom-associativity condition.

A morphism of BiHom-associative algebras is a linear map such that , , and .

A BiHom-associative algebra is called unital if there exists an element (called a unit) such that and

Definition 2 ([7]). A BiHom-coassociative coalgebra is a 4-tuple , in which is a linear space, and and are linear maps, such that , , , and

The maps and (in this order) are called the structure maps of , and condition (3) is called the BiHom-coassociativity condition.

Let us record the formula expressing the BiHom-coassociativity of :

A morphism of BiHom-coassociative coalgebras is a linear map such that , , and .

A BiHom-coassociative coalgebra is called counital if there exists a linear map (called a counit) such that

Definition 3 ([7]). A BiHom-bialgebra is a 7-tuple , with the property that is a BiHom-associative algebra, is a BiHom-coassociative coalgebra and moreover, the following relations are satisfied, for all :

We say that is a unital and counital BiHom-bialgebra if, in addition, it admits a unit and a counit such that

Let be a unital and counital BiHom-bialgebra with a unit and a counit . A linear map is called an antipode if it commutes with all the maps , and it satisfies the following relation:

A BiHom-Hopf algebra is a unital and counital BiHom-bialgebra with an antipode.

We can get some properties of the antipode. The proof is similar to the monoidal BiHom-Hopf algebra case in ([7], Proposition 6.6).

Proposition 4. Let be a BiHom-Hopf algebra. Then,

Definition 5 ([7, 10]).
Let be a BiHom-associative algebra and a triple where is a linear space, and are commuting linear maps. (i) is a left -module if we have a linear map , , such that , , andIf and are left -modules (both -actions denoted by ), a morphism of left -modules is a linear map satisfying the conditions and , for all and .
If is a unital BiHom-associative algebra and is a left -module, then is called unital if , for all . (ii) is a right -module if we have a linear map , , such that , , andIf is a unital BiHom-associative algebra and is a right -module, then is called unital if , for all . (iii)If is a left -module and a right -module, then is called an -bimodule if(iv)Let be a BiHom-bialgebra for which the maps are bijective. A BiHom-associative algebra is called a left -module BiHom-algebra if is a left -module, with action denoted by , such that the following condition is satisfied:

3. The Maschke-Type Theorem for the BiHom-Smash Product

In this section, we will give a Maschke-type theorem for the BiHom-smash product over a semisimple BiHom-Hopf algebra .

Definition 6 ([7]). Let be a BiHom-bialgebra and a left -module BiHom-algebra, with the left action such that all structure maps are bijective. The BiHom-smash product is defined on the vector space , and the BiHom-multiplication is given by for all . Note that is a BiHom-associative algebra. Moreover, if and are both unital with the units and , then also has a unit .

We assume that the BiHom-smash product in our paper is unital.

Proposition 7. Let be a BiHom-smash product, then there are two BiHom-algebra isomorphisms via and via . This means and , for all and .

Proof. A straightforward computation left to the reader.

Our next result is the BiHom-analogue of the integral (for a Hom-analogue, see [17] and monoidal Hom-analogue, see [16]).

Definition 8. Let be a BiHom-Hopf algebra. A left integral in is an element which is and -invariant (i.e., ) such that for all . A left integral is normalized if . Similarly, we can define right integrals in . We denote the space of left and right integrals in by and . If , then we say is unimodular. is semisimple if and only if possesses a normalized left integral if and only if possesses a normalized right integral.

Proposition 9. Let be a finite-dimensional BiHom-Hopf algebra with , be an -module BiHom-algebra, and , , be left -modules. If is an -module map, then is an -module morphism, where .

Proof. First, show that . For all , Similarly, from , we get .
Since is finite-dimensional and there is a nonzero integral , it follows that is bijective. Meanwhile, all structure maps are bijective; thus, for , there exists an element such that . For any , we have which implies that is a left -module morphism. Furthermore, we have for all . Meanwhile, obtains by the following computation: for all . Next, we claim that is a left -module morphism. Indeed, for all , we get Thus, we get that is a left -module morphism.

Proposition 10. Let be a finite-dimensional semisimple BiHom-Hopf algebra and be a left -module BiHom-algebra. Let be a left -module and an -submodule of . If is a direct summand of as -modules, then is also a direct summand of as -modules.

Proof. Since is a finite-dimensional semisimple BiHom-Hopf algebra, there exists a normalized right integral in . Let be a canonical projection as -modules. Define Thus, by Proposition 9, we obtain that is a left -module morphism. Now, we show that is also a projection. By the projrctivity of , we prove for all that It follows which is a direct summand of as left -modules.
By the above discussions, we obtain the Maschke-type theorem for BiHom-smash product, which generalizes Theorem 14 in [16].

Theorem 11. Let be a finite-dimensional semisimple BiHom-Hopf algebra and be a left -module BiHom-algebra. If is semisimple, then so is the BiHom-smash product .

4. The Associated Morita Context

The main aim of this section is to construct an associated Morita context between the BiHom-smash product and the BiHom-subalgebra . Note that the monoidal Hom-analogue of the Morita context has been studied in [16].

Let be a left -module BiHom-algebra and be the -invariants of . It is easy to get . If , then both and are in . It follows which is a BiHom-subalgebra of . For all , we only check

Let be a normalized left integral of ; then for all , we have

This means . Conversely, if , then and . So, . In summary, if is a BiHom-Hopf algebra with a normalized left integral , then .

Lemma 12. Let be a BiHom-Hopf algebra with a bijective antipode and be a left -module BiHom-algebra. Then, is a left and right -module via the following module structure maps: (i)(ii)for all .

Proof. We prove the conditions (10) and (11) hold and leave the others to the reader. For (10), For (11), for all and .

Lemma 13. Let be a BiHom-Hopf algebra with a bijective antipode and be a left -module BiHom-algebra. From Lemma 12, is a left and right -module. Meanwhile, it is also a left and right -module with the BiHom-multiplication. Thus, there are two bimodules and .

Proof. By the BiHom-associativity, it is easy to get which is a left and right -module. Now, we check the condition (12) hold for and as follows: for all and .
Let be a left integral in . If , then is also a right integral in . For all , For any , So, we obtain Now, with the above preparations, we can construct the associated Morita context between the BiHom-smash product and the BiHom-subalgebra .

Theorem 14. Let be a BiHom-Hopf algebra with a bijective antipode , be a left integral in satisfying , and be a left -module BiHom-algebra. Then, forms an associated Morita context with the maps for all .

Proof. We first prove is both an -module map and middle -linear map, which means satisfies the following three conditions: for all and .
To prove these conditions, we compute Next, since , thus the map is well defined. We now prove that is both an -module map and middle -linear. For any , we compute Finally, the BiHom-associativity is obtained by We get forms an associated Morita context.

Data Availability

There is no data available.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors sincerely thank the referee for his or her valuable suggestions and comments on this paper. This work was supported by the NSF of China (Nos. 11801515 and 12071441), the Natural Science Foundation of Zhejiang Province (No. LY20A010003), the Foundation of Zhejiang Educational Committee (No. Y201942625), and the Project of Zhejiang College, Shanghai University of Finance and Economics (No. 2020YJYB01).