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

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                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <mfenced open="(" close=")">
                           <mrow>
                              <mi>H</mi>
                              <mo>,</mo>
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                                 <mrow>
                                    <mi>α</mi>
                                 </mrow>
                                 <mrow>
                                    <mi>H</mi>
                                 </mrow>
                              </msub>
                              <mo>,</mo>
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                                 <mrow>
                                    <mi>β</mi>
                                 </mrow>
                                 <mrow>
                                    <mi>H</mi>
                                 </mrow>
                              </msub>
                              <mo>,</mo>
                              <msub>
                                 <mrow>
                                    <mi>ω</mi>
                                 </mrow>
                                 <mrow>
                                    <mi>H</mi>
                                 </mrow>
                              </msub>
                              <mo>,</mo>
                              <msub>
                                 <mrow>
                                    <mi>ψ</mi>
                                 </mrow>
                                 <mrow>
                                    <mi>H</mi>
                                 </mrow>
                              </msub>
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                                    <mi>S</mi>
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                                    <mi>H</mi>
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                  </jats:inline-formula> be a BiHom-Hopf algebra and <jats:inline-formula>
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                           <mrow>
                              <mi>A</mi>
                              <mo>,</mo>
                              <msub>
                                 <mrow>
                                    <mi>α</mi>
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                                 <mrow>
                                    <mi>A</mi>
                                 </mrow>
                              </msub>
                              <mo>,</mo>
                              <msub>
                                 <mrow>
                                    <mi>β</mi>
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                                    <mi>A</mi>
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                     </math>
                  </jats:inline-formula> be an <jats:inline-formula>
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                        <mfenced open="(" close=")">
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                              <mi>H</mi>
                              <mo>,</mo>
                              <msub>
                                 <mrow>
                                    <mi>α</mi>
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                                 <mrow>
                                    <mi>H</mi>
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                              <mo>,</mo>
                              <msub>
                                 <mrow>
                                    <mi>β</mi>
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                                    <mi>H</mi>
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                  </jats:inline-formula>-module BiHom-algebra. Then, in this paper, we study some properties on the BiHom-smash product <jats:inline-formula>
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                  </jats:inline-formula>. We construct the Maschke-type theorem for the BiHom-smash product <jats:inline-formula>
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                  </jats:inline-formula> and form an associated Morita context <jats:inline-formula>
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                  </jats:inline-formula>.</jats:p>


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, Hombialgebras, 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, BiHomassociative 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][9][10][11][12].
Let H 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 BiHomsmash 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 A#H. If ðH, μ H , Δ H , α H , β H , ψ H , ω H Þ is a finite-dimensional semisimple BiHom-Hopf algebra, then we construct the Maschke-type A BiHom-associative algebra ðA, μ, α, βÞ is called unital if there exists an element 1 A ∈ A (called a unit) such that Definition 2 ([7]). A BiHom-coassociative coalgebra is a 4tuple ðC, Δ, ψ, ωÞ, in which C is a linear space, and ψ, ω : C ⟶ C and Δ : The maps ψ and ω (in this order) are called the structure maps of C, and condition (3) is called the BiHomcoassociativity condition.
We can get some properties of the antipode. The proof is similar to the monoidal BiHom-Hopf algebra case in ( [7], Proposition 6.6).

The Maschke-Type Theorem for the BiHom-Smash Product A#H
In this section, we will give a Maschke-type theorem for the BiHom-smash product ðA#H, β H Þ is defined on the vector space A ⊗ H, and the BiHommultiplication is given by for all a, a′ ∈ A, h, h′ ∈ H. Note that ðA#H, We assume that the BiHom-smash product in our paper is unital.
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]).
is an ðA#H,

Advances in Mathematical Physics
which implies thatλ is a left ðH, α H , β H Þ-module morphism. Furthermore, we have for all a ∈ A, h ∈ H. Meanwhile, obtains by the following computation: for all a ∈ A, h ∈ H. Next, we claim thatλ is a left ðA, α A , β A Þ -module morphism. Indeed, for all a ∈ A, m ∈ M, we get Advances in Mathematical Physics Thus, we get thatλ is a left ðA#H, α A ⊗ α H , β A ⊗ β H Þ -module morphism.
Thus, by Proposition 9, we obtain thatλ is a left ðA#H, α A ⊗ α H , β A ⊗ β H Þ-module morphism. Now, we show thatλ is also a projection. By the projrctivity of λ, we prove for all n ∈ N that ð24Þ By the above discussions, we obtain the Maschke-type theorem for BiHom-smash product, which generalizes Theorem 14 in [16].

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