On Hom-Lie Pseudo-Superalgebras

The aim of this article is to introduce the notion of Hom-Lie H-pseudo-superalgebras for any Hopf algebra H. This class of algebras is a natural generalization of the Hom-Lie pseudo-algebras as well as a special case of the Hom-Lie superalgebras. We present some construction theorems of HomLie H-pseudo-superalgebras, reformulate the equivalent definition of Hom-Lie H-pseudo-superalgebras, and consider the cohomology theory of Hom-Lie H-pseudo-superalgebras with coefficients in arbitrary Hom-modules as a generalization of Kac’s result.


Introduction
The notion of conformal algebras [1]- [5] was introduced by Kac as a formal language describing the singular part of the operator product expansion in two-dimensional conformal field theory, and it came to be useful for investigation of vertex algebras (see [6]- [8]).The concept of vertex algebras was derived from mathematical physics; it was first mathematically defined and considered by Borcherds in [9] to obtain his solution of the Moonshine conjecture in the theory of finite simple groups.
As a generalization of conformal algebras, Bakalov, D'Andrea and Kac [10] developed a theory of "multidimensional" lie conformal algebras, called Lie H-pseudo-algebras for any Hopf algebra H. Classification problems, cohomology theory and representation theory have been considered in [10]- [12].In [13], Boyallian and Liberati studied pseudo-algebras from the point of view of pseudo-dual of classical Lie coalgebra structures by defining the notions of Lie H-coalgebras and Lie pseudo-bialgebras.
In [14], Sun generalized the pseudo-algebra structures to the Hom-pseudo-algebras of associative and Lie type, and showed some examples of the new structures and construction theorems.Hom-algebras were firstly studied by Hartwig, Larsson and Silvestrov in [15], where they introduced the structure of Hom-Lie algebras in the context of the deformations of Witt and Virasoro algebras.Later, Larsson and Silvestrov extended the notion of Hom-Lie algebras to quasi-Hom Lie algebras and quasi-Lie algebras (see [16]).Recently, Yau laid the foundation of a homology theory for Hom-Lie algebras and constructed the enveloping algebras of Hom-Lie and Hom-Leibniz algebras in [17]- [19].Many more properties and structures of Hom-Lie algebras have been developed (see [20]- [23] and references cited therein).
In [24], Hom-algebras and Hom-coalgebras were introduced by Makhlouf and Silvestrov as a generalization of ordinary algebras and coalgebras in the following sense: the associativity of the multiplication was replaced by the Hom-associativity and similar for Hom-coassociativity.They also defined the structures of Hom-bialgebras and Hom-Hopf algebras, and described some of their properties extending properties of ordinary bialgebras and Hopf algebras in [25] and [26].Different to Makhlouf and Silvestrov's work, Caenepeel and Goyvaerts studied the Hom-Hopf algebras from a categorical view point in [27], and called them monoidal Hom-bialgebras and monoidal Hom-Hopf algebras respectively (for more details about monoidal Hom-Hopf algebras, see references [28]- [32] and references cited therein).
In [33], Ammar and Makhlouf introduced the notion of Hom-Lie superalgebras and provided a construction theorem from which one can derive a one parameter family of Hom-Lie superalgebras deforming the orthosymplectic Lie superalgebras.The notion of Hom-Lie superalgebras is a natural and meaningful generalization of Lie superalgebras which were introduced by Kac in [3].Motivated by [4] [10], in which Kac formulated the notion of conformal superalgebras and considered the classification theorem and representation theory of conformal superalgebras.We think whether we can extend the notions of Hom-Lie pseudo-algebras and conformal superalgebras to Hom-Lie pseudo-superalgebras.
Cohomology is an important tool in mathematics.Its range of applications contains algebra and topology as well as the theory of smooth manifolds or of holomorphic functions.The cohomology theory of Lie algebras was developed by Chevalley, Eilenberg and Cartan.Scheunert and Zhang introduced and investigated the cohomology groups of Lie superalgebras in [34].Naturally, we think whether we can extend the notion of cohomology groups to Hom-Lie H-pseudo-superalgebras.This becomes our second motivation of the paper.
To give a positive answer to the questions above, we organize this paper as follows.In Section 2, we recall some basic definitions about Lie pseudo-algebras.In Section 3, we define Hom-Lie pseudo-superalgebras and introduce two construction theorems of Hom-Lie pseudo-superalgebras (see Proposition 3.12 and Theorem 3.13).In Section 4, we mainly discuss the annihilation superalgebras of Hom-pseudo-superalgebras (see Proposition 4.5).In Section 5, we determine some equivalent definitions of Hom-pseudo-superalgebras.In Section 6, we discuss the cohomology of Hom-Lie H-pseudo-superalgebras (see Theorem 6.1).

Preliminaries
In this section we recall some basic definitions and results related to our paper.Throughout the paper, all algebraic systems are supposed to be over a field k of characteristic 0, H always denotes a Hopf algebra with an antipode S. We summarize in the following the ungraded definitions of Hom-associative and Hom-Lie H-pseudoalgebras (see [14]).The reader is referred to Sweedler [35] about Hopf algebras, the Sweedler-type notation for the comultiplication is denoted by: ( ) ( ) ( ) is a category whose objects are the same objects as in the category H M of left H-modules, but with a non-trivial pseudotensor structure, see [10].
A Hom-associative H-pseudo-algebra [14] is a triple ( ) , 2) Hom-Jacobi identity: An elementary but important property of Hom-Lie H-pseudo-algebra is that each Hom-associative H-pseudoalgebra gives rise to a Hom-Lie H-pseudo-algebra via the commutator bracket.
3) Hom-Jacobi identity: Recall from Sun [14] we know that one can reformulate the definition of a Hom-Lie H-pseudo-algebra via a Hom-Lie H-conformal algebra.

Hom-Pseudo Superalgebras of Associative and Lie Types
In this section we will introduce the concept and construction theorems of Hom-H-pseudo-superalgebras of associative and Lie types, and show some examples of Hom-Lie H-pseudo-superalgebras that are neither Hom-Lie superalgebras nor Hom-Lie pseudo-algebras.A Hom-associative H-pseudo-superalgebra ( ) , , , then the Hom-associative H-pseudo-superalgebra ( ) , , A µ α is multiplicative., .

Let ( ) , ,
A α * be a Hom-associative H-pseudo-superalgebra and M be a superapace in . , Hom M M and satisfies the following properties ( , , where Example 3.4.Let ( ) , A α be a finite dimensional Hom-associative superalgebra, H be a Hopf algebra.Then ( ) 2) Hom-Jacobi identity: where a, b, c are homogeneous elements in L.
Here and further, a is the parity of a.  * is noting but the affine Hom-Lie superalgebra in [33].
A Hom-Lie H-pseudo-superalgebra , then the Hom-Lie H-pseudo-superalgebra ( ) , , , . , Hom M M and satisfies the following axioms: where and m are homogeneous elements in L and M respectively.In the following, we will show that the supercommutator bracket defined using the multiplication in a Homassociative H-pseudo-superalgebra leads naturally to a Hom-Lie H-pseudo-superalgebra.Lemma 3.11.Let ( ) Proof.We only prove (3), and similarly for (1), (2).For any homogeneous elements , , , .
,  , ,  A α * is a Hom-Lie H-pseudo-superalgebra.Proof.We shall show that the condition (3.9) leads A to be a Hom-Lie H-pseudo-superalgebra, in the sense of Definition 3.5.For this purpose, we first claim that the bracket pseudoproduct is both H-bilinear and skewcommutative, but these are easy to check.It remains to verify that the conditions (2) of Definition 3.5 are satisfied by the condition (3.9).Now we have the following calculations: Immediately, we can obtain It follows from Lemma 3.12 that .
Together with the above results, we finally obtain The proof is completed. Next we will construct Hom-Lie H-pseudo-superalgebras from Lie H-pseudo-superalgebras and even Hom-Lie superalgebra endomorphisms, generalizing the results for Hom-Lie H-pseudo-algebras in [14] and Hom-Lie superalgebras in [33]., is a morphism of Hom-Lie H-pseudo-superalgebras.
Proof.We shall show that [ ] ( ) L α α * satisfies the skew-commutativity and the Hom-Jacobi identity.For any homogeneous elements , , .
as needed.To show that f is a morphism of Hom-Lie H-pseudo-superalgebras, we do the calculations: .
The proof is completed. To provides another way to construct Hom-Lie H-pseudo-superalgebras and Hom-associative H-pseudosuperalgebras, we first recall the definition of current H-pseudo-algebras in [10].
Let H ′ be a Hopf subalgebra of H and A an H'-pseudo-algebra.Then define the current H-pseudo-algebra

Then ( )
, , is a Hom-Lie H-pseudo-superalgebra, where γ is the multiplication of CurL.
Moreover, there is a similar result in the case of Hom-associative H'-pseudo-superalgebras as well.
Proof.We only prove the case of Hom-Lie H'-pseudo-superalgebras, the Hom-associative case is similar.We denote It is obviously that the skew-commutativity holds since is a Hom-Lie H'-pseudo-superalgebra.So it is sufficient to verify the Hom-Jacobi identity.For any , , is a Hom-Lie H'-pseudo-superalgebra, we have , .
By the multiplication of CurA , we obtain .
CurL γ β is a Hom-Lie H-pseudo-superalgebra.This ends the proof.

Hom-Annihilation Superalgebras
In this section we will study the annihilation superalgebras of Hom-H-pseudo-superalgebras.First of all we will give the definition of H-differential superalgebras.
, , , for all h H ∈ and , , where [ ] First we shall show that A Y L is an H-module, but this is easy to check.It remains to verify that the conditions (1) and (2) in Definition 3.5 are satisfied.For this purpose, we take , , as required.Next we verify the Hom-Jacobi identity by the following calculations: .
is a Hom-Lie H-differential superalgebra, we call it Hom-annihilation superalgebra of the Hom-Lie H-pseudo-algebra L and write x H a x a = ⊗ for any homogeneous elements a L ∈ and .x X ∈ Remark 4.4.A similar statement holds for Hom-associative H-pseudo-superalgebras and Hom-modules as well.For example, if ( ) , , , for any homogeneous elements , .
According to the bracket pseudoproduct defined above, it is easy to see that H-bilinearity holds.To verify the Skew-commutativity and Hom-Jacobi identity, take , , , , , .
) ( ) That is, the skew-commutativity holds.So it is sufficient to verify the Hom-Jacobi identity.Since , .
Similarly, by exchanging the status of the element , , ( ) .
So A Y L is a Hom-Lie H-pseudo-superalgebra.This completes the proof.

Hom-Lie Conformal Superalgebras
In this section we will reformulate the definition of Hom-Lie (or Hom-associative) H-pseudo-superalgebras.The resulting notion, equivalent to that of Hom-H-pseudo-superalgebras, will be called Hom-H-conformal superalgebras.
Let us start by racalling the definitions of the Fourier transform and the x-brackets in [10].For an arbitrary Hopf algebra H, the Fourier transform : F is an isomorphism with an inverse given by ( The significance of Fourier transform F is the identity [ ] ( Then for x X H * ∈ = , the x-bracket is defined in [3] as follows: [ ] ( be a Hom-Lie H-pseudo-superalgebra.For any homogeneous elements , , Then we have , , , .
where i x and i y are dual bases of X and H, a, b, c are homogeneous elements in L, h H ∈ and , x y X ∈ .In the following we will show that there is a simple relationship between the x-bracket of a Hom-Lie H-conformal superalgebra and the commutator in its annihilation Hom-Lie H-pseudo-superalgebra ( ) According to Proposition 4.2, we obtain , .
Below we give one way of constructing Hom-modules over Hom-Lie H-pseudo-algebras, whose proofs are similar to that in [10].
, , if , for all homogeneous elements , a L x X ∈ ∈ and m M ∈ .This action is compatible with the action of H, that is, where i h and i x are dual linear basis of H and X.

Cohomology of Hom-Lie H-Pseudo-Superalgebras
In this section, we will consider the cohomology of Hom-Lie H-pseudo-superalgebras, generalizing the results of Hom-Lie H-pseudoalgebras and Lie superalgebras.
Explicitly, γ has the following defining properties: ) 2) Skew-supersymmetry: For any , where , 1 : is the transposition of the ith and ( ) The map γ is called even (resp.odd) when ( ) ) for all even (resp.odd) elements where 1 i σ → is the permutation  The fact that 2 0 d = is most easily checked and the same argument is in the usual Lie superalgebra case in [26] [36] [37] and Hom-Lie H-pseudoalgebra case in [34].The cohomology of the resulting complex γ ∈ and define an action *     by (6.2).Then *     depends only on the γ .

α
consisting of a linear space L in H M , an oper-

Definition 3 . 1 . 2 .
A Hom-associative H-pseudo-superalgebra is a triple ( ) For a one dimensional Hopf algebra H = k, a Hom-associative H-pseudo-superalgebra is just a Hom-associative superalgebra over k.If id α = , then a Hom-associative H-pseudo-superalgebra is an associa- tive H-pseudo-superalgebra.

Proposition 3 . 12 .
as required.So (3) holds since A is Hom-associative.Given any Hom-associative H-pseudo-superalgebra ( ) be a Lie H-pseudo-superalgebra and α an even endomorphisms of L. Defining pseudoproduct a b * of A using the H-bilinearity.Explicitly, for any , a b A ∈ , define -Lie H-pseudo-superalgebra and Y a commutative associative H-differential superalgebra with a right action of H. Then

1 )
In order to reformulate the definition of a Lie (or associative) H-pseudo-algebra, Bakalov, D'Andrea and Kac introduced the bracket [ ] , a b H L ∈ ⊗ as the Fourier transform of [ ] a b * :

Definition 5 . 1 .α
A Hom-Lie H-conformal superalgebra is a triple consisting of a superspace L in H M , an even operation [ ]

α
a, b, c are homogeneous elements in L and h H ∈ .One can also reformulate Definition 4.1 in terms of x-brackets [ ] x a b as below.Definition 5.2.A Hom-Lie H-conformal superalgebra is a triple consisting of a superspace L in H M , an even operation [ ]:

• 8 )
One can also modify the above definition by replacing everywhere H ⊗ by ⊗ .Let the reduced cohomology.In the following we will show that the cohomology theory of Hom-Lie H-pseudo-superalgebras describes extensions and deformations, just as any cohomology theory.Hom-Lie H-pseudo-superalgebra, and ( ) -module, considering a Hom-Lie H-pseudo-superalgebra with respect to the zero pseudobracket, then the equivalence classes of H-split abelian extensions the homogeneous component of degree zero of the reduced cohomology() → be an extension of L-modules, which is split over H. Choosing a splitting{ } ˆ| , L L M l m l L m M = ⊕ = + ∈ ∈as an H-module, and denoting the pseudobracket of L by â bIt is not hard to verify that γ is a homogeneous 2-cochain of degree zero, i.e., super Jacobi identity of L and L implies 0 dγ = in the sense of (6.1).Conversely, given an element of algebra which is Lie or associative when A is so.