Received 19 March 2016; accepted 16 May 2016; published 19 May 2016
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1. 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 “multi- dimensional” 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 [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).
2. 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-pseudo- algebras (see [14] ). The reader is referred to Sweedler [35] about Hopf algebras, the Sweedler-type notation for the comultiplication is denoted by:
.
Recall that a pseudotensor category
is a category whose objects are the same objects as in the category
of left H-modules, but with a non-trivial pseudotensor structure, see [10] .
A Hom-associative H-pseudo-algebra [14] is a triple
consisting of a linear space A in
, an operation
and a homomorphism
satisfying
(2.1)
A Hom-Lie H-pseudo-algebra [14] is a triple
consisting of a linear space L in
, an operation
and a homomorphism
satisfying the following axioms (
):
1) Skew-commutativity:
(2.2)
2) Hom-Jacobi identity:
(2.3)
An elementary but important property of Hom-Lie H-pseudo-algebra is that each Hom-associative H-pseudo- algebra gives rise to a Hom-Lie H-pseudo-algebra via the commutator bracket.
A Hom-Lie H-conformal algebra ( [14] ) is a triple
consisting of a linear space L in
, an operation
and a homomorphism
satisfying the following axioms (
and
):
1) H-sesqui-linearity:
(2.4)
2) Skew-commutativity:
(2.5)
3) Hom-Jacobi identity:
(2.6)
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.
3. 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.
Definition 3.1. A Hom-associative H-pseudo-superalgebra is a triple
consisting of a superspace A in
, an even operation
and an even homomorphism
satisfying
(3.1)
in
for all homogeneous elements ![]()
Example 3.2. For a one dimensional Hopf algebra H = k, a Hom-associative H-pseudo-superalgebra is just a Hom-associative superalgebra over k. If
, then a Hom-associative H-pseudo-superalgebra is an associative H-pseudo-superalgebra.
A Hom-associative H-pseudo-superalgebra
is called multiplicative if
. For example, if
, then the Hom-associative H-pseudo-superalgebra
is multiplicative.
Let
and
be two (multiplicative) Hom-associative H-pseudo-superalgebras, an even homomorphism
is said to be a morphism of Hom-associative H-pseudo-superalgebras if
(3.2)
Definition 3.3. Let
be a Hom-associative H-pseudo-superalgebra and M be a superapace in
A Hom-A-module is a triple
, where
is an even morphism in
,
is an even morphism in
and satisfies the following properties (
):
(3.3)
(3.4)
where
.
Example 3.4. Let
be a finite dimensional Hom-associative superalgebra, H be a Hopf algebra. Then
is a Hom-associative H-pseudo-superalgebra with pseudoproduct
given by
![]()
for all
and homogeneous elements ![]()
Definition 3.5. A Hom-Lie H-pseudo-superalgebra is a triple
consisting of a superspace L in
, an even operation
and an even homomorphism
satisfying the following axioms:
1) Skew-commutativity:
(3.5)
2) Hom-Jacobi identity:
(3.6)
where a, b, c are homogeneous elements in L.
Here and further,
is the parity of a.
Example 3.6. For a one dimensional Hopf algebra H = k, a Hom-Lie H-pseudo-superalgebra is just a Hom-Lie superalgebra over k. If
, then a Hom-Lie H-pseudo-superalgebra is a Lie H-pseudo-superalgebra.
Example 3.7. Let H be a Hopf algebra and
a 2-dimensional linear superspace, where
is generated by x and
is generated by y. Then
is a Hom-Lie H-pseudo-superalgebra, where
is a free pseudo-algebra of rank 2 with pseudoproduct given by
in
is any even homomorphism in ![]()
Example 3.8. Let
be a finite dimensional Hom-Lie superalgebra, H be a Hopf algebra. Then
is a Hom-Lie H-pseudo-superalgebra with pseudoproduct
given by
![]()
for all
and homogeneous elements ![]()
Example 3.9. Let H be a Hopf algebra and
a 3-dimensional linear superspace, where A0 is generated by x, y and A1 is generated by z. Then
is a Hom-Lie H-pseudo-superalgebra defined by any even homomorphism
and operation
![]()
![]()
In particular, if
, then the Hom-Lie H-pseudo-superalgebra
is noting but the affine Hom-Lie superalgebra in [33] .
A Hom-Lie H-pseudo-superalgebra
is called multiplicative if
. For example, if
, then the Hom-Lie H-pseudo-superalgebra
is multiplicative.
Let
and
be two (multiplicative) Hom-Lie H-pseudo-superalgebras. An even homomorphism
is said to be a morphism of Hom-Lie H-pseudo-superalgebras if
(3.7)
Definition 3.10. Let
be a Hom-Lie H-pseudo-superalgebra and M a superspace in
A Hom-L-module is a triple
, where
is an even morphism in
,
is an even morphism in
and satisfies the following axioms:
(3.8)
where
, a, b 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 Hom- associative H-pseudo-superalgebra leads naturally to a Hom-Lie H-pseudo-superalgebra.
Lemma 3.11. Let
be a Hom-associative H-pseudo-superalgebra. Then
1) ![]()
2) ![]()
3) ![]()
![]()
Proof. We only prove (3), and similarly for (1), (2). For any homogeneous elements
let
![]()
![]()
On one hand we have
![]()
since H is cocommutative. Similarly, we have
![]()
as required. So (3) holds since A is Hom-associative. ,
Proposition 3.12. Given any Hom-associative H-pseudo-superalgebra
, one can define the bracket pseudoproduct on homogeneous elements by
(3.9)
and then extending by linearity to all elements. Then
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 skew- commutative, 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
, then
![]()
It follows from Lemma 3.12 that
![]()
Furthermore, we have
![]()
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] .
Theorem 3.13. Let
be a Lie H-pseudo-superalgebra and
an even endomorphisms of L. Defining
by
for all homogeneous elements x, y in L, then
is a Hom-Lie H-pseudo-superalgebra.
Moreover, suppose that
is another Lie H-pseudo-superalgebra and
is an even endomorphisms of
. If
is a morphism of Lie H-pseudo-superalgebras that satisfies
, then
(3.10)
is a morphism of Hom-Lie H-pseudo-superalgebras.
Proof. We shall show that
satisfies the skew-commutativity and the Hom-Jacobi identity. For any homogeneous elements
in L,
![]()
Since
is an endomorphism of L,
![]()
Therefore we have
![]()
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-pseudo- superalgebras, we first recall the definition of current H-pseudo-algebras in [10] .
Let
be a Hopf subalgebra of H and A an H'-pseudo-algebra. Then define the current H-pseudo-algebra
by extending the pseudoproduct
of A using the H-bilinearity. Explicitly, for any
, define
(3.11)
if
. Then
is an H-pseudo-algebra which is Lie or associative when A is so.
Proposition 3.14. Let H' be a Hopf subalgebra of H and
a Hom-Lie H'-pseudo-superalgebra. Then
is a Hom-Lie H-pseudo-superalgebra, where
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
(3.12)
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
, suppose
![]()
![]()
![]()
Since
is a Hom-Lie H'-pseudo-superalgebra, we have
![]()
that is,
![]()
By the multiplication of
, we obtain
![]()
Hence
is a Hom-Lie H-pseudo-superalgebra. This ends the proof. ,
4. 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.
Definition 4.1. An associative superalgebra Y is called an associative H-differential superalgebra if it is a left H-module such that
, for all
and homogeneous elements
.
Let Y be an H-bimodule which is a commutative associative H-differential superalgebra. For a left H-module L, it is easy to see that
is a left H-module via
, for all
and
.
The definition of Hom-Lie H-differential-superalgebras can be obtained similarly.
Proposition 4.2. Let Y be a Hom-Lie H-differential-superalgebra and
a Hom-Lie H-pseudo- superalgebra. Then AYL is a Hom-Lie H-differential superalgebra, where the bracket and the action are given by
(4.1)
(4.2)
for all
and
, where
.
Proof. First we shall show that AYL 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
, and suppose
![]()
![]()
![]()
Since L is a Hom-Lie H-pseudo-superalgebra, then
therefore we have
![]()
as required. Next we verify the Hom-Jacobi identity by the following calculations:
![]()
Similarly, by exchanging the status of the element
, we have
![]()
![]()
By the Hom-Jacobi identity of L, we have
![]()
Hence
![]()
So AYL is a Hom-Lie H-differential superalgebra. This completes the proof. ,
Remark 4.3. In particular, when
,
is a Hom-Lie H-differential superalgebra, we call it Hom-annihilation superalgebra of the Hom-Lie H-pseudo-algebra L and write
for any homogeneous elements
and ![]()
Remark 4.4. A similar statement holds for Hom-associative H-pseudo-superalgebras and Hom-modules as well. For example, if
is a Hom-L-module, then
is a Hom-AYL- module with a compatible H-action, where
(4.3)
if
for any homogeneous elements
and ![]()
Proposition 4.5. Let
be a Hom-Lie H-pseudo-superalgebra and Y a commutative associative H-differential superalgebra with a right action of H. Then
is a Hom-Lie H-pseudo- superalgebra with bracket pseudoproduct
(4.4)
if
for any homogeneous elements ![]()
Proof. 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
and suppose
![]()
![]()
![]()
Since L is a Hom-Lie H-pseudo-superalgebra,
therefore we have
![]()
That is, the skew-commutativity holds. So it is sufficient to verify the Hom-Jacobi identity. Since
![]()
we have
![]()
Similarly, by exchanging the status of the element
, we have
![]()
![]()
By the Hom-Jacobi identity of L, we have
![]()
it follows that
![]()
So AYL is a Hom-Lie H-pseudo-superalgebra. This completes the proof. ,
5. 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
is defined by
F is an isomorphism with an inverse given by
The significance of Fourier transform F is the identity
(5.1)
In order to reformulate the definition of a Lie (or associative) H-pseudo-algebra, Bakalov, D'Andrea and Kac introduced the bracket
as the Fourier transform of
:
![]()
That is,
![]()
Then for
, the x-bracket is defined in [3] as follows:
![]()
Let
be a Hom-Lie H-pseudo-superalgebra. For any homogeneous elements
, suppose
![]()
![]()
![]()
Then we have
![]()
![]()
Similarly, we can obtain
thus
![]()
![]()
Therefore
![]()
is equivalent to
![]()
So the definition of Hom-Lie H-pseudo-superalgebra can be equivalently reformulated as follows.
Definition 5.1. A Hom-Lie H-conformal superalgebra is a triple
consisting of a superspace L in
, an even operation
and an even homomorphism
satisfying the following axioms:
1) H-sesqui-linearity:
(5.2)
2) Skew-commutativity:
(5.3)
3) Hom-Jacobi identity:
(5.4)
where a, b, c are homogeneous elements in L and
.
One can also reformulate Definition 4.1 in terms of x-brackets
as below.
Definition 5.2. A Hom-Lie H-conformal superalgebra is a triple
consisting of a superspace L in
, an even operation
and an even homomorphism
satisfying the following axioms:
1) Locality:
(5.5)
2) H-sesqui-linearity:
(5.6)
3) Skew-super commutativity:
(5.7)
4) Hom-super Jacobi identity:
(5.8)
where
and
are dual bases of X and H, a, b, c are homogeneous elements in L,
and
.
In the following we will show that there is a simple relationship between the x-bracket of a Hom-Lie H-con- formal superalgebra and the commutator in its annihilation Hom-Lie H-pseudo-superalgebra
defined in Proposition 4.5. Let
be dual linear basis of H and X. Then we have
![]()
According to Proposition 4.2, we obtain
![]()
In other words,
![]()
Below we give one way of constructing Hom-modules over Hom-Lie H-pseudo-algebras, whose proofs are similar to that in [10] .
Proposition 5.3. Any Hom-module
over a Hom-Lie H-pseudo-superalgebra
has a natural structure of a Hom-A(L)-module, given by
, where
(5.9)
for all homogeneous elements
and
. This action is compatible with the action of H, that is,
for all homogeneous elements
and
, and satisfies the locality
condition:
for any homogeneous elements
and
.
Conversely, any Hom-A(L)-module
satisfying the above conditions has a natural structure of an Hom-L-module, given by
(5.10)
where
and
are dual linear basis of H and X.
6. 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.
Let
be a Hom-Lie H-pseudo-superalgebra,
is a Hom-L-module. Let
be a natural number and let
be the superspace of all homogeneous skew-symmetric cochains
satisfies
(6.1)
Explicitly,
has the following defining properties:
1) H-polylinearity: For any
and
,
(6.2)
2) Skew-supersymmetry: For any ![]()
(6.3)
where
is the transposition of the ith and
st factors.
The map
is called even (resp. odd) when
(resp.
) for all even (resp. odd) elements
, where the parity of the element
is
We denote the parity of the map
by
.
For
, the map
is defined as follows:
(6.4)
where
is the permutation
,
is the permutation
and the sign ^ indicates that the element below it must be omitted. In particular, for
we have
(6.5)
and for
we obtain
(6.6)
The fact that
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
is called the cohomology of
with coefficients in
and is denoted by ![]()
One can also modify the above definition by replacing everywhere
by
. Let
consist of all skew-symmetric cochains
. Then we can define a differential
by (6.1) with
replaced by
everywhere; then again
The corresponding cohomology
will be called the basic cohomology of
with coefficients in
. In contrast,
is sometimes called 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.
Theorem 6.1. Let
be a multiplicative Hom-Lie H-pseudo-superalgebra, and
be a Hom-L-module, considering a Hom-Lie H-pseudo-superalgebra with respect to the zero pseudobracket, then the equivalence classes of H-split abelian extensions
(6.7)
of the Hom-Lie H-pseudo-superalgebra
correspond bijectively to
, the homogeneous component of degree zero of the reduced cohomology
.
Proof. Let
be an extension of L-modules, which is split over H. Choosing a splitting
as an H-module, and denoting the pseudobracket of
by
, we have for all
:
(6.8)
It is not hard to verify that
is a homogeneous 2-cochain of degree zero, i.e.,
The Hom- super Jacobi identity of L and
implies
in the sense of (6.1).
Conversely, given an element of
, we can choose a representative
and define an action
by (6.2). Then
depends only on the
. ,
Acknowledgements
The paper is partially supported by the Project Funded by China Postdoctoral Science Foundation (No. 2015M571725), the Key University Science Research Project of Anhui Province (Nos. KJ2015A294 and KJ2014A183) and the NSF of Chuzhou University (Nos. 2015qd01, 2014qd008 and 2014PY08).
NOTES
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*Corresponding author.