# The Maschke-Type Theorem and Morita Context for BiHom-Smash Products

Authors: Bingliang Shen; Ling Liu
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

LetH,αH,βH,ωH,ψH,SH be a BiHom-Hopf algebra and A,αA,βA be an H,αH,βH-module BiHom-algebra. Then, in this paper, we study some properties on the BiHom-smash product A#H. We construct the Maschke-type theorem for the BiHom-smash product A#H and form an associated Morita context AH,AHAA#H,A#HAAH,A#H.


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## Body

## 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].LetH be a Hopf algebra and A an H-module algebra; then, as well known, we can construct the smash product algebra A#H (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 Section2, 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 A#H. If H,μH,ΔH,αH,βH,ψH,ωH is a finite-dimensional semisimple BiHom-Hopf algebra, then we construct the Maschke-type theorem for the BiHom-smash product A#H. We also prove that AH,AHAA#H,A#HAAH,A#H forms an associated Morita context, where AH,αAAH,βAAH is a BiHom-subalgebra of H,αH,βH-invariants in A,αA,βA.


## 2. Preliminaries

We work over a base fieldk. All algebras, linear spaces, etc. will be over k; unadorned ⊗ means ⊗k. We use Sweedler’s notation for terminologies on coalgebras. For a coalgebra C, we write comultiplication Δc=c1⊗c2, for any c∈C.Definition 1 ([7]).
A BiHom-associative algebra is a 4-tupleA,μ,α,β, where A is a linear space and α,β:A⟶A and μ:A⊗A⟶A are linear maps such that α∘β=β∘α, αx·y=αx·αy, βx·y=βx·βy and
(1)αx·y·z=x·y·βz,for all x,y,z∈A. The maps α and β (in this order) are called the structure maps of A, and condition (1) is called the BiHom-associativity condition.A morphismf:A,μA,αA,βA⟶B,μB,αB,βB of BiHom-associative algebras is a linear map f:A⟶B such that αB∘f=f∘αA, βB∘f=f∘βA, and f∘μA=μB∘f⊗f.A BiHom-associative algebraA,μ,α,β is called unital if there exists an element 1A∈A (called a unit) such that α1A=1A,β1A=1A and
(2)a1A=αaand1Aa=βa,∀a∈A.Definition 2 ([7]).
A BiHom-coassociative coalgebra is a 4-tupleC,Δ,ψ,ω, in which C is a linear space, and ψ,ω:C⟶C and Δ:C⟶C⊗C are linear maps, such that ψ∘ω=ω∘ψ, ψ⊗ψ∘Δ=Δ∘ψ, ω⊗ω∘Δ=Δ∘ω, and
(3)Δ⊗ψ∘Δ=ω⊗Δ∘Δ.The mapsψ and ω (in this order) are called the structure maps of C, and condition (3) is called the BiHom-coassociativity condition.Let us record the formula expressing the BiHom-coassociativity ofΔ:
(4)Δc1⊗ψc2=ωc1⊗Δc2,∀c∈C.A morphismg:C,ΔC,ψC,ωC⟶D,ΔD,ψD,ωD of BiHom-coassociative coalgebras is a linear map g:C⟶D such that ψD∘g=g∘ψC, ωD∘g=g∘ωC, and g⊗g∘ΔC=ΔD∘g.A BiHom-coassociative coalgebraC,Δ,ψ,ω is called counital if there exists a linear map ε:C⟶k (called a counit) such that
(5)ε∘ψ=ε,ε∘ω=ε,idC⊗ε∘Δ=ωandε⊗idC∘Δ=ψ.Definition 3 ([7]).
A BiHom-bialgebra is a 7-tupleH,μ,Δ,α,β,ψ,ω, with the property that H,μ,α,β is a BiHom-associative algebra, H,Δ,ψ,ω is a BiHom-coassociative coalgebra and moreover, the following relations are satisfied, for all h,h′∈H:
(6)Δhh′=h1h′1⊗h2h′2,α∘ψ=ψ∘α,α∘ω=ω∘α,β∘ψ=ψ∘β,β∘ω=ω∘β,α⊗α∘Δ=Δ∘α,β⊗β∘Δ=Δ∘β,ψhh′=ψhψh′,ωhh′=ωhωh′.We say thatH is a unital and counital BiHom-bialgebra if, in addition, it admits a unit 1H and a counit εH such that
(7)Δ1H=1H⊗1H,εH1H=1,ψ1H=1H,ω1H=1H,εH∘α=εH,εH∘β=εH,εHhh′=εHhεHh′,∀h,h′∈H.LetH,μ,Δ,α,β,ψ,ω be a unital and counital BiHom-bialgebra with a unit 1H and a counit εH. A linear map S:H⟶H is called an antipode if it commutes with all the maps α,β,ψ,ω, and it satisfies the following relation:
(8)βψSh1αωh2=εHh1H=βψh1αωSh2,∀h∈H.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.
LetH,μ,Δ,α,β,ψ,ω,S be a BiHom-Hopf algebra. Then,
(9)S1H=1H,εH∘S=εH,Sβaαb=SβbSαa,∀a,b∈H,ψSh1⊗ωSh2=ωSh2⊗ψSh1,∀h∈H.Definition 5 ([7, 10]).
LetA,μA,αA,βA be a BiHom-associative algebra and M,αM,βM a triple where M is a linear space, and αM,βM:M⟶Mare commuting linear maps.
(i)
M,αM,βM is a left A-module if we have a linear map A⊗M⟶M, a⊗m↦a·m, such that αMa·m=αAa·αMm, βMa·m=βAa·βMm, and(10)αAa·a′·m=a·a′·βMm,∀a,a′∈A,m∈M.
IfM,αM,βM and N,αN,βN are left A-modules (both A-actions denoted by ·), a morphism of left A-modules f:M⟶N is a linear map satisfying the conditions αN∘f=f∘αM,βN∘f=f∘βM and fa·m=a·fm, for all a∈A and m∈M.
IfA,μA,αA,βA,1A is a unital BiHom-associative algebra and M,αM,βM is a left A-module, then M is called unital if 1A·m=βMm, for all m∈M.
(ii)
M,αM,βM is a right A-module if we have a linear map M⊗A⟶M, m⊗a↦m·a, such that αMm·a=αMm·αAa, βMm·a=βMm·βAa, and(11)αMm·a·a′=m·a·βAa′,∀a,a′∈A,m∈M.
IfA,μA,αA,βA,1A is a unital BiHom-associative algebra and M,αM,βM is a right A-module, then M is called unital if m·1A=αMm, for all m∈M.
(iii)
IfM,αM,βM is a left A,αA,βA-module and a right B,αB,βB-module, then M is called an A,B-bimodule if(12)αAa·m·a′=a·m·βBa′,∀a∈A,a′∈B,m∈M.(iv)
LetH,μH,ΔH,αH,βH,ψH,ωH be a BiHom-bialgebra for which the maps αH,βH,ψH,ωH are bijective. A BiHom-associative algebra A,μA,αA,βA is called a left H-module BiHom-algebra if A,αA,βA is a left H,αH,βH-module, with action denoted by H⊗A⟶A,h⊗a↦h·a, such that the following condition is satisfied:(13)h·aa′=αH−1ωH−1h1·aβH−1ψH−1h2·a′,∀h∈H,a,a′∈A.


## 3. The Maschke-Type Theorem for the BiHom-Smash ProductA#H

In this section, we will give a Maschke-type theorem for the BiHom-smash productA#H,αA⊗αH,βA⊗βH over a semisimple BiHom-Hopf algebra H,μH,ΔH,αH,βH,ψH,ωH,SH.Definition 6 ([7]).
LetH,μH,ΔH,αH,βH,ψH,ωH be a BiHom-bialgebra and A,μA,αA,βA a left H-module BiHom-algebra, with the left action H⊗A⟶A,h⊗a↦h·a such that all structure maps αH,βH,ψH,ωH,αA,βA are bijective. The BiHom-smash product A#H,αA⊗αH,βA⊗βH is defined on the vector space A⊗H, and the BiHom-multiplication is given by
(14)a#ha′#h′=aβH−1ωH−1h1·βA−1a′#ψH−1h2h′,for all a,a′∈A,h,h′∈H. Note that A#H,αA⊗αH,βA⊗βH is a BiHom-associative algebra. Moreover, if A and H are both unital with the units 1A and 1H, then A#H,αA⊗αH,βA⊗βH also has a unit 1A#1H.We assume that the BiHom-smash product in our paper is unital.Proposition 7.
LetA#H,αA⊗αH,βA⊗βH be a BiHom-smash product, then there are two BiHom-algebra isomorphisms A≅A#1H via a↦a#1H and H≅1A#H via h↦1A#h. This means a#1Hb#1H=ab#1H and 1A#h1A#g=1A#hg, for all a,b∈A and h,g∈H.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.
LetH,μH,ΔH,αH,βH,ψH,ωH,1H,εH,SH be a BiHom-Hopf algebra. A left integral in H is an element t∈H which is αH and βH-invariant (i.e., αHt=t,βHt=t) such that
(15)ht=εht,for all h∈H. A left integral t is normalized if εHt=1. Similarly, we can define right integrals in H. We denote the space of left and right integrals in H by ∫lH and ∫rH. If ∫lH=∫rH, then we say H is unimodular. H is semisimple if and only if H possesses a normalized left integral if and only if H possesses a normalized right integral.Proposition 9.
LetH,μH,ΔH,αH,βH,ψH,ωH,1H,εH,SH be a finite-dimensional BiHom-Hopf algebra with 0≠t∈∫rH, A,αA,βA be an H,αH,βH-module BiHom-algebra, and M,αM,βM, N,αN,βN, be left A#H,αA⊗αH,βA⊗βH-modules. If λ:M,αM,βM⟶N,αN,βN is an A,αA,βA-module map, then
(16)λ~:M,αM,βM⟶N,αN,βN,m↦βHωH−1SHt1·λψH−1t2·βM−2mis an A#H,αA⊗αH,βA⊗βH-module morphism, where ΔHt=t1⊗t2.Proof.
First, show thatαN∘λ~=λ~∘αM. For all m∈M,
(17)αN∘λ~m=αNλ~m=αHβHωH−1SHt1·αNλψH−1t2·βM−2m=βHωH−1SHαHt1·λψH−1αHt2·αMβM−2m=βHωH−1SHt1·λψH−1t2·βM−2αMmbyαHt=t=λ~αMm.
Similarly, fromβHt=t, we get βNλ~m=λ~βMm.
SinceH,αH,βH,ψH,ωH is finite-dimensional and there is a nonzero integral t, it follows that SH is bijective. Meanwhile, all structure maps αH,βH,ψH,ωH,αA,βA are bijective; thus, for g∈H, there exists an element h∈H such that g=SHαHβH3ωHψH2h. For any m∈M, we have
(18)g·λ~m=SHαHβH3ωHψH2h·βHωH−1SHt1·λψH−1t2·βM−2m=7SHβH3ωHψH2hSHβHωH−1t1·λβHψH−1t2·βM−1m=SHβHβH2ωHψH2hSHαHωH−1t1·λαHψH−1t2·βM−1mbyαHt=t=75SHβHωH−1t1αHβH2ωHψH2h·λαHψH−1t2·βM−1m=SHβHωH−1t1αHβH2ψH2h1εHh2·λαHψH−1t2·βM−1m=SHβHωH−1t1αHβH2ψH2h1·λψH−1t21HεHh2·βM−1m=753SHαHβHωH−1t1αHβH2ψH2h1·λαHψH−1t2βHψHh21αHωHSHh22·βM−1m=7531SHαHβHωH−1t1αHβH2ψH2h1·λψH−1t2βHψHh21αHβHωHSHh22·βM−1m=75312SHαHβHωH−1t1αHβH2ωH−1ψH2h11·λψH−1t2βHψHh12αHβHωHψHSHh2·βM−1m=SHαHβHωH−1t1βHψH2h11·λψH−1t2βHψH2h12αHβHωHψHSHh2·βM−1m=SHαHβHωH−1t1εHh1·λψH−1t2αHβHωHψHSHh2·βM−1m=SHαHβHωH−1t1·λψH−1t2αHβHωHψH2SHh·βM−1m=βHωH−1SHt1·λαH−1ψH−1t2βH−2g·βM−1m=753127βHωH−1SHt1·λψH−1t2·βH−2g·βM−2m=βHωH−1SHt1·λψH−1t2·βM−2g·m=λ~g·m,which implies that λ~ is a left H,αH,βH-module morphism. Furthermore, we have
(19)1A#ha#1H=ωH−1h1·a#αHψH−1h2,a#1H1A#h=αAa#βHh,for all a∈A,h∈H. Meanwhile,
(20)a#h=1A#αH−1ψH−1h2SH−1αH−1βH−1ωH−1h1·βA−2a#1Hobtains by the following computation:
(21)1A#αH−1ψH−1h2SH−1αH−1βH−1ωH−1h1·βA−2a#1H=2αH−1ψH−1ωH−1h21·SH−1αH−1βH−1ωH−1h1·βA−2a#ψH−2h22=27αH−2ψH−1ωH−1h21SH−1αH−1βH−1ωH−1h1·βA−1a#ψH−2h22=272αH−2ψH−1ωH−1h12SH−1αH−1βH−1ωH−2h11·βA−1a#ψH−1h2=SH−1βHSHαH−2βH−1ψH−1ωH−1h12SH−1αHαH−2βH−1ωH−2h11·βA−1a#ψH−1h2=2725SH−1αH−2ωH−2h11SHαH−1βH−1ψH−1ωH−1h12·βA−1a#ψH−1h2=SH−1βHψHαH−2βH−1ωH−2ψH−1h11SHαHωHαH−1βH−2ψH−1ωH−2h12·βA−1a#ψH−1h2=27253SH−11H·βA−1a#ψH−1h2εHh1=2725341H·βA−1a#ψH−1εHh1h2=a#h,for all a∈A,h∈H. Next, we claim that λ~ is a left A,αA,βA-module morphism. Indeed, for all a∈A,m∈M, we get
(22)a·λ~m=a·βHωH−1SHt1·λψH−1t2·βM−2m=a#1H·1A#βHωH−1SHt1·λψH−1t2·βM−2m=7αA−1a#1H1A#βHωH−1SHt1·λβHψH−1t2·βM−1m=73a#βH2ωH−1SHt1·λβHψH−1t2·βM−1m=a#βHωH−1SHt1·λψH−1t2·βM−1mbyβHt=t=7341A#αH−1βHωH−1ψH−1SHt12SH−1αH−1ωH−2SHt11·βA−2a#1H·λψH−1t2·βM−1m=73461A#αH−1βHωH−2SHt11αH−1ωH−1ψH−1t12·βA−2a#1H·λψH−1t2·βM−1m=1A#βHωH−2SHt11ωH−1ψH−1t12·βA−2a#1H·λαHψH−1t2·βM−1mbyαHt=t=734621A#βHωH−1SHt1ωH−1ψH−1t21·βA−2a#1H·λαHψH−2t22·βM−1m=7346271A#αHβHωH−1SHt1·ωH−1ψH−1t21·βA−2a#1H·λαHβH−1ψH−2t22·βM−2m=1A#αHβHωH−1SHt1·λωH−1ψH−1t21·βA−2a#1H·1A#αHβH−1ψH−2t22·βM−2m=73462771A#αHβHωH−1SHt1·λαH−1ωH−1ψH−1t21·αA−1βA−2a#1H1A#αHβH−1ψH−2t22·βM−1m=734627731A#αHβHωH−1SHt1·λωH−1ψH−1t21·βA−2a#αHψH−2t22·βM−1m=1A#αHβHωH−1SHt1·λωH−1ψH−1t21·βA−2a#αHψH−1ψH−1t22·βM−1m=7346277321A#αHβHωH−1SHt1·λ1A#ψH−1t2βA−2a#1H·βM−1m=73462773271A#αHβHωH−1SHt1·λ1A#αHψH−1t2·βA−2a#1H·βM−2m=1A#βHωH−1SHt1·λ1A#ψH−1t2·βA−2a·βM−2mbyαHt=t=βHωH−1SHt1·λψH−1t2·βM−2a·m=λ~a·m.
Thus, we get thatλ~ is a left A#H,αA⊗αH,βA⊗βH-module morphism.Proposition 10.
LetH,μH,ΔH,αH,βH,ψH,ωH,1H,εH,SH be a finite-dimensional semisimple BiHom-Hopf algebra and A,αA,βA be a left H,αH,βH-module BiHom-algebra. Let M,αM,βM be a left A#H,αA⊗αH,βA⊗βH-module and N,αN,βN an A#H,αA⊗αH,βA⊗βH-submodule of M,αM,βM. If N,αN,βN is a direct summand of M,αM,βM as A,αA,βA-modules, then N,αN,βN is also a direct summand of M,αM,βM as A#H,αA⊗αH,βA⊗βH-modules.Proof.
SinceH,μH,ΔH,αH,βH,ψH,ωH,1H,εH,SH is a finite-dimensional semisimple BiHom-Hopf algebra, there exists a normalized right integral t in H. Let λ:M,αM,βM⟶N,αN,βN be a canonical projection as A,αA,βA-modules. Define
(23)λ~:M,αM,βM⟶N,αN,βN,m↦βHωH−1SHt1·λψH−1t2·βM−2m.
Thus, by Proposition9, we obtain that λ~ is a left A#H,αA⊗αH,βA⊗βH-module morphism. Now, we show that λ~ is also a projection. By the projrctivity of λ, we prove for all n∈N that
(24)λ~n=βHωH−1SHt1·λψH−1t2·βN−2n=βHωH−1SHt1·ψH−1t2·βN−2n=7αH−1βHωH−1SHt1ψH−1t2·βN−1n=βHωH−1SHt1αHψH−1t2·βN−1nbyαHt=t=ωH−1ψH−1βHψHSHt1αHωHt2·βN−1n=73εHt1H·βN−1n=1H·βN−1n=n.
It followsN,αN,βN which is a direct summand of M,αM,βM as left A#H,αA⊗αH,βA⊗βH-modules.
By the above discussions, we obtain the Maschke-type theorem for BiHom-smash product, which generalizes Theorem 14 in [16].Theorem 11.
LetH,μH,ΔH,αH,βH,ψH,ωH,1H,εH,SH be a finite-dimensional semisimple BiHom-Hopf algebra and A,αA,βA be a left H,αH,βH-module BiHom-algebra. If A,αA,βA is semisimple, then so is the BiHom-smash product A#H,αA⊗αH,βA⊗βH.


## 4. The Associated Morita Context

The main aim of this section is to construct an associated Morita context between the BiHom-smash productA#H,αA⊗αH,βA⊗βH and the BiHom-subalgebra AH,αAAH,βAAH. Note that the monoidal Hom-analogue of the Morita context has been studied in [16].LetA,αA,βA be a left H,αH,βH-module BiHom-algebra and
(25)AH=a∈A∣h·a=εhβAa,∀h∈Hbe the H,αH,βH-invariants of A,αA,βA. It is easy to get 1A∈AH. If a,a′∈AH, then both αAa,βAa and aa′ are in AH. It follows AH,αAAH,βAAH which is a BiHom-subalgebra of A,αA,βA. For all a,a′∈AH,h∈H, we only check
(26)h·aa′=10αH−1ωH−1h1·aβH−1ψH−1h2·a′=εHh1βAaεHh2βAa′=εHhβAaa′.Lett be a normalized left integral of H,αH,βH; then for all h∈H,a∈A, we have
(27)h·t·a=7αH−1ht·βAa=εHht·βAa=εHhβHt·βAa=εHhβHt·a.This meanst·a∈AH. Conversely, if a∈AH, then βA−1a∈AH and t·βA−1a=εHta=a. So, AH⊂t·A. In summary, if H,μH,ΔH,αH,βH,ψH,ωH,1H,εH,SH is a BiHom-Hopf algebra with a normalized left integral t, then t·A=AH.Lemma 12.
LetH,μH,ΔH,αH,βH,ψH,ωH be a BiHom-Hopf algebra with a bijective antipode SH and A,αA,βA be a left H,αH,βH-module BiHom-algebra. Then, A,αA,βA is a left and right A#H,αA⊗αH,βA⊗βH-module via the following module structure maps:
(i)
a#h⟶b=aβA−1h·b,(ii)
b⟵a#h=SH−1αH−1βHh·βA−1ba,
for alla,b∈A,h∈H.Proof.
We prove the conditions (10) and (11) hold and leave the others to the reader. For (10),
(28)a#hb#k⟶βAc=1aβH−1ωH−1h1·βA−1b#ψH−1h2k⟶βAc=aβH−1ωH−1h1·βA−1bβA−1ψH−1h2k·βAc=aβH−1ωH−1h1·βA−1bβH−1ψH−1h2βH−1k·c=17αAaβH−1ωH−1h1·βA−1bβH−2ψH−1h2βH−2k·βA−1c=177αAaβH−1ωH−1h1·βA−1bαHβH−2ψH−1h2·βH−2k·βA−2c=αAaαH−1ωH−1αHβH−1h1·βA−1bβH−1ψH−1αHβH−1h2·βH−2k·βA−2c=17710αAaαHβH−1h·βA−1bβH−2k·βA−2c=αAaβA−1αHh·bβA−1k·c=αAa#αHh⟶bβA−1k·c=αAa#αHh⟶b#k⟶c.
For (11),
(29)αAa⟵b#hc#k=1αAa⟵bβH−1ωH−1h1·βA−1c#ψH−1h2k=SH−1αH−1βHψH−1h2k·βA−1αAabβH−1ωH−1h1·βA−1c=17αH−1βHSH−1βHβH−1ψH−1h2αHαH−1k·βA−1abωH−1h1·c=175αH−1βHSH−1βHαH−1kαH−1βHSH−1αHβH−1ψH−1h2·βA−1abωH−1h1·c=αH−2βH2SH−1kSH−1ψH−1h2·βA−1abωH−1h1·c=1757αH−1βH2SH−1k·SH−1ψH−1h2·βA−2abωH−1h1·c=175710αH−1βH2SH−1k·αH−1ωH−1ψH−1SH−1h21·βA−2abβH−1ψH−1ψH−1SH−1h22·βH−2ωH−1h1·βA−2c=1757106αH−1βH2SH−1k·αH−1ψH−2SH−1h22·βA−2abβH−1ωH−1ψH−1SH−1h21·βH−2ωH−1h1·βA−2c=17571067αH−1βH2SH−1k·αH−1ψH−2SH−1h22·βA−2abαH−1βH−1ωH−1ψH−1SH−1h21βH−2ωH−1h1·βA−1c=175710672αH−1βH2SH−1k·αH−1ψH−1SH−1h2·βA−2abαH−1βH−1ωH−1ψH−1SH−1h12βH−2ωH−2h11·βA−1c=αH−1βH2SH−1k·αH−1ψH−1SH−1h2·βA−2abSH−1βHαH−1βH−2ωH−1ψH−1h12SH−1αHαH−1βH−2ωH−2SHh11·βA−1c=1757106725αH−1βH2SH−1k·αH−1ψH−1SH−1h2·βA−2abSH−1αH−1βH−1ωH−2SHh11βH−2ωH−1ψH−1h12·βA−1c=αH−1βH2SH−1k·αH−1ψH−1SH−1h2·βA−2abSH−1βHψHαH−1βH−2ωH−2ψH−1SHh11αHωHαH−1βH−2ωH−2ψH−1h12·βA−1c=17571067253αH−1βH2SH−1k·αH−1ψH−1SH−1h2·βA−2abεHh11H·βA−1c=αH−1βH2SH−1k·αH−1SH−1h·βA−2abc=SH−1αH−1βHβHk·βA−1αH−1SH−1βHh·βA−1abβAc=SH−1αH−1βHh·βA−1ab⟵βAc#βHk=a⟵b#h⟵βAc#βHk,for all a,b,c∈A and h,k∈H.Lemma 13.
LetH,μH,ΔH,αH,βH,ψH,ωH be a BiHom-Hopf algebra with a bijective antipode SH and A,αA,βA be a left H,αH,βH-module BiHom-algebra. From Lemma 12, A,αA,βA is a left and right A#H,αA⊗αH,βA⊗βH-module. Meanwhile, it is also a left and right AH,αAAH,βAAH-module with the BiHom-multiplication. Thus, there are two bimodules AHAA#H,αA,βA and A#HAAH,αA,βA.Proof.
By the BiHom-associativity, it is easy to getA,αA,βA which is a left and right AH,αAAH,βAAH-module. Now, we check the condition (12) hold for AHAA#H,αA,βA and A#HAAH,αA,βA as follows:
(30)ab⟵βAc#βHh=SH−1αH−1βH2h·βA−1abβAc=7SH−1αH−1βH2h·βA−1αAabc=SH−1αH−1βH2h·αAβA−1aβA−1bc=710αH−1ωH−1SH−1αH−1βH2h1·αAβA−1aβH−1ψH−1SH−1αH−1βH2h2·βA−1bc=7106αH−2βH−2ψH−1SH−1h2·αAβA−1aαH−1βHωH−1SH−1h1·βA−1bc=εHh2αAaαH−1βHωH−1SH−1h1·βA−1bc=αAaSH−1αH−1βHh·βA−1bc=αAab⟵c#h,αAc#αHh⟶ba=αAcβA−1αHh·ba=10αAcβA−1ωH−1h1·bαHβH−1ψH−1h2·a=αAcβA−1ωH−1h1·bεHh2βAa=αAcβA−1h·ba=107cβA−1h·bβAa=c#h⟶bβAa,for all a∈AH,b,c∈A and h∈H.
Lett be a left integral in H,αH,βH,ψH,ωH. If SHt=t, then t is also a right integral in H,αH,βH,ψH,ωH. For all t∈H,
(31)th=SHtSHSH−1h=SHβHtSHαHαH−1SH−1h=5SHβHαH−1SH−1hSHαHt=αH−1βHhSHt=αH−1βHht=εHht.
For anya∈A,h∈H,
(32)1A#ta#h=41A#t1A#αH−1ψH−1h2SH−1αH−1βH−1ωH−1h1·βA−2a#1H=471A#t1A#αH−1ψH−1h2SH−1αH−1ωH−1h1·βA−1a#1H=1A#tαH−1ψH−1h2SH−1αH−1ωH−1h1·βA−1a#1H=1A#tεHh2SH−1αH−1ωH−1h1·βA−1a#1H=1A#tSH−1αH−1h·βA−1a#1H.
So, we obtain(33)1A#ta#h=1A#tSH−1αH−1h·βA−1a#1H.
Now, with the above preparations, we can construct the associated Morita context between the BiHom-smash productA#H and the BiHom-subalgebra AH.Theorem 14.
LetH,αH,βH,ψH,ωH be a BiHom-Hopf algebra with a bijective antipode SH, t be a left integral in H satisfying SHt=t, and A,αA,βA be a left H,αH,βH-module BiHom-algebra. Then, AH,AHAA#H,A#HAAH,A#H forms an associated Morita context with the maps
(34)−,−:A⊗AHA,αA⊗αA,βA⊗βA⟶A#H,αA⊗αH,βA⊗βH,viaa,b=a#tb#1H,−,−:A⊗A#HA,αA⊗αA,βA⊗βA⟶AH,αAAH,βAAH,viaa,b=t·βA−1ab,for all a,b∈A.Proof.
We first prove−,− is both an A#H,αA⊗αH,βA⊗βH-module map and middle AH,αAAH,βAAH-linear map, which means −,− satisfies the following three conditions:
(35)αAa#αHhb,c=a#h⟶b,βAc,a,bβAc#βHh=αAa,b⟵c#h,aa′,βAb=αAa,a′b,for all a,b,c∈A,h∈H and a′∈AH.
To prove these conditions, we compute(36)αAa#αHhb,c=αAa#αHhb#tc#1H=1a#hb#tβAc#1H=13αA−1a#1H1A#βH−1hαA−1b#1H1A#tβAc#1H=131a#1H1A#βH−1hαA−1βA−1b#1H1A#tβAc#1H=1311a#1H1A#αH−1βH−1hαA−1βA−1b#1H1A#tβAc#1H=13112a#1HαH−1βH−1ωH−1h1·αA−1βA−1b#βH−1ψH−1h21A#tβAc#1H=131123a#1HαH−2βH−1ωH−1h1·αA−2βA−1b#1H1A#βH−2ψH−1h21A#tβAc#1H=1311231a#1HαH−1βH−1ωH−1h1·αA−1βA−1b#1H1A#βH−2ψH−1h21A#tβAc#1H=a#1HαH−1βH−1ωH−1h1·αA−1βA−1b#1H1A#βH−2ψH−1h2tβAc#1H=a#1HαH−1βH−1ωH−1h1·αA−1βA−1b#1H1A#εHh2tβAc#1H=a#1HαH−1βH−1h·αA−1βA−1b#1H1A#tβAc#1H=13112313a#1HβH−1h·βA−1b#tβAc#1H=a1H·βA−1βA−1h·b#tβAc#1H=aβA−1h·b#tβAc#1H=aβA−1h·b,βAc=a#h⟶b,βAc,a,bβAc#βHh=a#tb#1HβAc#βHh=3αA−1a#1H1A#tb#1HαA−1βAc#1H1A#h=31a#1H1A#tβA−1b#1HαA−1βAc#1H1A#h=311αAa#1H1A#tβA−1b#1HαA−1c#1H1A#βH−1h=3111αAa#1H1A#tβA−1b#1HαA−1βA−1c#1H1A#βH−2h=31111αAa#1H1A#tαA−1βA−1b#1HαA−1βA−1c#1H1A#βH−1h=αAa#1H1A#tαA−1βA−1bc#1H1A#βH−1h=311113αAa#1H1A#tβA−1bc#h=3111131αAa#1H1A#tSH−1αH−1h·βA−2bc#1H=31111311a#1H1A#tSH−1αH−1βHh·βA−1bc#1H=αAa#tSH−1αH−1βHh·βA−1bc#1H=αAa,SH−1αH−1βHh·βA−1bc=αAa,b⟵c#hαAa,a′b=αAa#ta′b#1H=αAa#ta′#1Hb#1H=1a#ta′#1HβAb#1H=aβH−1ωH−1t1·βA−1a′#ψH−1t21HβAb#1H=aεHt1a′#ψH−1t21HβAb#1H=aa′#t1HβAb#1H=aa′#tβAb#1H=aa′,βAb.
Next, sinceAH=t·A, thus the map −,− is well defined. We now prove that −,− is both an AH,αAAH,βAAH-module map and middle A#H,αA⊗αH,βA⊗βH-linear. For any a,b∈A,a′∈AH,h∈H, we compute
(37)a′a,βAb=t·βA−1a′aβAb=1t·βA−1αAa′ab=t·αAβA−1a′βA−1ab=110αH−1ωH−1t1·αAβA−1a′βH−1ψH−1t2·βA−1ab=εHt1αAa′βH−1ψH−1t2·βA−1ab=αAa′βH−1t·βA−1ab=αAa′t·βA−1ab=αAa′a,b,αAa,ba′=t·βA−1αAaba′=1t·βA−1abβAa′=t·βA−1aba′=110αH−1ωH−1t1·βA−1abβH−1ψH−1t2·a′=αH−1ωH−1t1·βA−1abεHt2βAa′=αH−1t·βA−1abβAa′=t·βA−1abβAa′=a,bβAa′,a⟵c#h,βAb=SH−1αH−1βHh·βA−1ac,βAb=t·βA−1SH−1αH−1βHh·βA−1acβAb=t·SH−1αH−1h·βA−2acb=t·SH−1αH−1ψH−1h2·βA−2acεHh1b=3t·SH−1αH−1ψH−1h2·βA−2acSH−1αH−1βH−1ωH−1ψH−1h12βH−2ωH−2h11·βA−1b=32t·SH−1αH−1ψH−2h22·βA−2acSH−1αH−1βH−1ωH−1ψH−1h21βH−2ωH−1h1·βA−1b=327t·SH−1αH−1ψH−2h22·βA−2acSH−1βH−1ωH−1ψH−1h21·βH−2ωH−1h1·βA−2b=3276t·αH−1ωH−1SH−1ψH−1h21·βA−2acβH−1ψH−1SH−1ψH−1h22·βH−2ωH−1h1·βA−2b=327610t·SH−1ψH−1h2·βA−2acβH−2ωH−1h1·βA−2b=3276107tSH−1ψH−1h2·βA−1acβH−1ωH−1h1·βA−1b=t·βA−1acβH−1ωH−1h1εHh2·βA−1b=t·βA−1acβH−1h·βA−1b=t·βA−1ach·b=32761077t·βA−1αAacβA−1h·b=αAa,cβA−1h·b=αAa,c#h⟶b.
Finally, the BiHom-associativity is obtained by(38)a,b⟶βAc=a#tb#1H⟶βAc=7αAa#t⟶b#1H⟶c=αAa#t⟶bβA−11H·c=αAa#t⟶bc=αAaβA−1t·bc=αAat·βA−1bc=αAab,cαAa⟵b,c=αAa⟵b#tc#1H=7a⟵b#t⟵βAc#1H=SH−1αH−1βHt·βA−1ab⟵βAc#1H=t·βA−1ab⟵βAc#1H=1H·βA−1t·βA−1abβAc=t·βA−1abβAc=a,bβAc.
We getAH,AHAA#H,A#HAAH,A#H forms an associated Morita context.



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*Source: 6677332-2021-01-15.xml*