
TL;DR
This paper introduces Hom-Lie antialgebras, explores their representations and cohomology, and establishes foundational results on extensions, deformations, and Nijenhuis operators for this new algebraic structure.
Contribution
It is the first to define Hom-Lie antialgebras and study their cohomology, extensions, deformations, and Nijenhuis operators, expanding the theory of Hom-algebra structures.
Findings
Equivalent classes of abelian extensions correspond to second cohomology groups.
1-parameter infinitesimal deformations are characterized by 2-cocycles.
Nijenhuis operators describe trivial deformations.
Abstract
In this paper, we introduced the notion of Hom-Lie antialgebras. The representations and cohomology theory of Hom-Lie antialgebras are investigated. We prove that the equivalent classes of abelian extensions of Hom-Lie antialgebras are in one-to-one correspondence to elements of the second cohomology group. We also prove that 1-parameter infinitesimal deformation of a Hom-Lie antialgebra are characterized by 2-cocycles of this Hom-Lie antialgebra with adjoint representation in itself. The notion of Nijenhuis operators of Hom-Lie antialgebra is introduced to describe trivial deformations.
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On Hom-Lie antialgebra
Tao Zhang
College of Mathematics and Information Science
Henan Normal University
Xinxiang 453007, PR China
and
Heyu Zhang
College of Mathematics and Information Science
Henan Normal University
Xinxiang 453007, PR China
Abstract.
In this paper, we introduced the notion of Hom-Lie antialgebras. The representations and cohomology theory of Hom-Lie antialgebras are investigated. We prove that the equivalent classes of abelian extensions of Hom-Lie antialgebras are in one-to-one correspondence to elements of the second cohomology group. We also prove that 1-parameter infinitesimal deformation of a Hom-Lie antialgebra are characterized by 2-cocycles of this Hom-Lie antialgebra with adjoint representation in itself. The notion of Nijenhuis operators of Hom-Lie antialgebra is introduced to describe trivial deformations.
Key words and phrases:
Hom-Lie antialgebra; cohomology; abelian extensions; deformations; Nijenhuis operators
2010 Mathematics Subject Classification:
Primary 17D99; Secondary 18G60
1. Introduction
The notion of Lie antialgebras was introduced by Ovsienko in [12]. A Lie antialgebra is a -graded vector space where is a commutative associative algebra, is equipped with a map satisfying some axioms similar to the axioms of Lie algebras, and acts commutatively on as a derivation, see Definition 2.1. The representations, universal enveloping algebra and cohomology theory of Lie antialgebras have been investigated in [7], [11] and [8].
On the other hand, Hom-type algebras was introduced to deal with -deformations of algebras of vector fields [5]. A Hom-associative algebra is a vector space with an additional linear map satisfying the Hom-associative identity:
[TABLE]
for all . A Hom-Lie algebra is a vector space with an additional linear map satisfying the Hom-Jacobi identity:
[TABLE]
for all . The representations, abelian extensions, deformations and cohomology theory of Hom-algebras were studied in [1, 10, 13]. Universal central extensions of Hom-Lie algebras were studied in [3, 9]. It is known that abelian extensions and deformations of Hom-type algebras are governed by the second cohomology group. Other types of Hom-structures include BiHom-Lie algebras, Hom-Nambu-Lie algebras, Hom-Hopf algebras, Hom-Poisson algebras and Hom-Lie-Yamaguti algebras, see [4, 6, 14, 15, 16].
Motivated by the above results, we introduce the notion of a Hom-Lie antialgebra in this paper. We define a Hom-Lie antialgebra as a supercommutative -graded algebra with two linear maps satisfying some compatibility conditions, see Definition 2.3. When both and are identity maps, we get the ordinary notion of a Lie antialgebra. Note that and are not equal to each other since they act on different spaces. This is the key difference of Hom-Lie antialgebra in this paper and other types of Hom-algebras in the literature. The representations and cohomology groups of Hom-Lie antialgebras are investigated. We also study deformations and abelian extensions of Hom-Lie antialgebras which are described by the second cohomology group. For the third cohomology group, it will be related to the crossed module extensions of a Hom-Lie antialgebra. This is investigated in our subsequent paper [17].
The paper is organized as follows. In Section 2, we give some definitions and notations of Hom-Lie antialgebras. In Section 3, we study representations of Hom-Lie antialgebras and define the cohomology groups of Hom-Lie antialgebras. In Section 4, we study abelian extensions of Hom-Lie antialgebras using the cohomology theory defined in Section 3 and prove that abelian extensions are classified by the second cohomology group. In Section 5, we study 1-parameter infinitesimal deformations of a Hom-Lie antialgebra. The notion of a Nijenhuis operator on a Hom-Lie antialgebra is introduced to describe trivial deformations.
Throughout this paper, we work with an algebraically closed field of characteristic 0. For a -graded vector space , we consider the standard -grading on the algebra of linear maps on : where and .
2. Hom-Lie antialgebras
Definition 2.1**.**
A Lie antialgebra is a supercommutative -graded algebra: , such that the following identities hold:
[TABLE]
for all homogeneous elements . Note that we denote by in the last equation (2.4) since it is anti-commutative, which is slightly different from notations in [12, 8].
Definition 2.2**.**
Let and be Lie antialgebras. An algebraic homomorphism from to consists of , , such that the following conditions hold:
[TABLE]
for all .
Definition 2.3**.**
A Hom-Lie antialgebra is a supercommutative -graded algebra , together with two linear maps , satisfying the following identities:
[TABLE]
for all . We also call such systems -Hom-Lie antialgebra.
We give some explanations about the meaning of equalities (2.8)–(2.11). From equality (2.8), is a Hom-associative subalgebra of . The equality (2.9) and (2.10) mean that is an action of on as derivations. A Hom-Lie antialgebra is called abelian if the product and bracket are all zero for any .
A Hom-Lie antialgebra is called multiplicative if form an algebraic homomorphism of , i.e. for any , we have
[TABLE]
The Hom-Lie antialgebras in this paper are assumed to be multiplicative unless otherwise stated.
Definition 2.4**.**
Let and be Hom-Lie antialgebras. A Hom-Lie antialgebra homomorphism from to consists of , , such that the following equalities hold for all :
[TABLE]
Proposition 2.5**.**
Let be a Lie antialgebra and be an algebraic homomorphism from to itself. Then the induced Hom-Lie antialgebra is the space under the following operations:
[TABLE]
Proof.
Here we verify that (2.9) and (2.10) hold, the other two are similar.
For (2.9),
[TABLE]
[TABLE]
thus, we have
[TABLE]
For (2.10),
[TABLE]
thus, we obtain
[TABLE]
∎
By the above Proposition 2.5, we can construct examples of Hom-Lie antialgebras as follows. More general constructions are given in the next sections.
Example 2.6**.**
Consider the Lie antialgebra introduced in [12] as follows. This algebra has the basis , where is even and are odd, satisfying the relations
[TABLE]
Consider the linear map defined by
[TABLE]
on the basis elements. This map is actually a Lie antialgebra homomorphism. By Proposition 2.5, we obtain a Hom-Lie algebra structure given by
[TABLE]
Example 2.7**.**
Another example of a Hom-Lie antialgebra is the conformal Hom-Lie antialgebra. This is a simple infinite-dimensional Hom-Lie antialgebra with the basis
[TABLE]
where are even, are odd, and , satisfy the following relations:
[TABLE]
where . **
3. Representations and cohomology
In this section, we introduce the notion of a representation for the class of Hom-Lie antialgebras. Then we study the semidirect products and cohomology groups of Hom-Lie antialgebras.
Definition 3.1**.**
Let be a Hom-Lie antialgebra, be a Hom-super vector space (a super vector space with linear maps ). A representation of over the Hom-super vector space is a pair of linear maps such that the following conditions hold:
[TABLE]
for all , .
The above conditions seem very complicated at first glance. We give an equivalent condition as follows.
Proposition 3.2**.**
Let be a Hom-Lie antialgebra, be a Hom-super vector space. Then is a representation of over if and only if is a Hom-Lie antialgebra under the following operations:
[TABLE]
for all , . This is called a semidirect product of and , denoted by .
The proof of the above proposition 3.2 is by direct computations, so we omit the details.
Now we define the generalized Chevalley-Eilenberg complex for Hom-Lie antialgebra with coefficients in . Given a Hom-Lie antialgebra with representation in , we define to be the space of multi-linear maps
[TABLE]
with and such that
[TABLE]
Denote by the set of -cochains:
[TABLE]
Define the coboundary operator , where for . The operator and are given explicitly as follows.
(i) If , is given by
[TABLE]
if , and is with values in , then is given by
[TABLE]
if , and is with values in , then is given by
[TABLE]
(ii) If and is odd, then is given by
[TABLE]
if or if is even, then is given by
[TABLE]
(iii) If , then is given by
[TABLE]
It can be proved similarly as in [8] that , so is a coboundary operator. Thus associated to the representation , we obtain the cochain complex \big{(}C^{k}(\mathfrak{a},V),d\big{)}. Denote the set of closed -cochains by and the set of exact -cochains by . Define the corresponding cohomology group by
[TABLE]
In particular, for , we obtain the following relations by direct computations:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
When in the above equations (3.33)–(3.36), is called a 2-cocycle.
4. Abelian extensions
In this section, we study abelian extensions of Hom-Lie antialgebra. It is proved that the equivalent classes of abelian extensions of Hom-Lie antialgebras are in one-to-one correspondence to the elements of the second cohomology group.
Definition 4.1**.**
Let , and be Hom-Lie antialgebras. An extension of by is a short exact sequence
[TABLE]
*of Hom-Lie antialgebras. It is called an abelian extension, if is an abelian ideal of , i.e. and . *
Definition 4.2**.**
Two extensions of Hom-Lie antialgebra
[TABLE]
and
[TABLE]
are called equivalent, if there exists a Hom-Lie antialgebra homomorphism such that , , that is to say the following diagram commutes
[TABLE]
We denote by the set of equivalence classes of extensions of by .
A section of consists of linear maps , such that , . Define the following maps
[TABLE]
by
[TABLE]
These two maps are well defined since is abelian. This gives a representation of on as the following lemma shows. The proof is routine, so we omit the details.
Lemma 4.3**.**
With above notations, is a representation of . Moreover, equivalent abelian extensions lead to the same representation.
Let be a section of an abelian extension. Define the following maps:
[TABLE]
for all and .
Lemma 4.4**.**
Let \textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathfrak{a}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0} be an abelian extension of by . Then defined by (4.40)–(4.42) is a -cocycle of with coefficients in , where the representation is given by (4.38)–(4.39).
Proof.
First, we prove that is a 2-cochain. Since and form an algebraic homomorphism of , we have
[TABLE]
Thus we get
[TABLE]
Similarly, one get
[TABLE]
Second, we prove that is a 2-cocycle. By the equality
[TABLE]
and equation (4.40), we have
[TABLE]
[TABLE]
Thus we obtain that
[TABLE]
Similarly, by the equality
[TABLE]
we obtain that
[TABLE]
By the equality
[TABLE]
we obtain that
[TABLE]
By the equality
[TABLE]
we obtain that
[TABLE]
When substituting , and with , and , one can see that the equations (4.43)–(4) correspond to (3.33)–(3.36), which are exactly the -cocycle conditions given at the end of last section. ∎
Now, we can obtain a Hom-Lie antialgebra structure on the space using the -cocycle given above.
Lemma 4.5**.**
Let be a Hom-Lie antialgebra, be an -module and is a -cocycle. Then is a Hom-Lie antialgebra under the following operations:
[TABLE]
where .
Proof.
We are going to check that with the above operations satisfies four axioms of Hom-Lie antialgebra. By direct computations, we have
[TABLE]
[TABLE]
Since is a Hom-Lie antialgebra and by the representation condition (3.20), we have
[TABLE]
Due to the 2-cocycle condition (4.43), we have
[TABLE]
Thus we obtain
[TABLE]
Analogously, by using the representation condition (3.21) and the 2-cocycle condition (4.44), one can prove the following equality:
[TABLE]
By direct computations, we also have
[TABLE]
[TABLE]
[TABLE]
Since is a Hom-Lie antialgebra and by the representation conditions (3.24) and (3.25), we have
[TABLE]
Due to the 2-cocycle condition (4.45), we have
[TABLE]
Thus we get
[TABLE]
Similarly, by the representation condition (3.26) and the 2-cocycle condition (4), we get
[TABLE]
Therefore, from equalities (4.47)–(4.50), we obtain that is a Hom-Lie antialgebra. The proof is completed. ∎
Lemma 4.6**.**
Two abelian extensions of Hom-Lie antialgebras
[TABLE]
and
[TABLE]
are equivalent if and only if and are in the same cohomology class.
Proof.
First, assume the above two abelian extensions are equivalent and be the corresponding homomorphism. As has to be the identity on , then there must exists a map such that
[TABLE]
where and .
Since is a homomorphism between Hom-Lie antialgebras and , we have
[TABLE]
The left hand side of (4.55) is equal to
[TABLE]
and the right hand side of (4.55) is equal to
[TABLE]
Thus we obtain
[TABLE]
By similar computations, we also obtain
[TABLE]
From equations (4.58)–(4.60), we obtain that and are in the same cohomology class.
Conversely, if and are in the same cohomology class, there exists a coboundary map such that . Then we can define the maps by (4.53) and (4.54). Similar as the above calculations, one can show that is an equivalence of the two abelian extensions. We omit the details. This finished the proof. ∎
From the above lemmas, we obtain the main theorem of this section.
Theorem 1**.**
Let be a Hom-Lie antialgebra and be a representation of over . Then there is a one-to-one correspondence between the set of equivalent classes of abelian extensions of the Hom-Lie antialgebra by and the elements in the second cohomology group .
5. Deformations
In this section, we study infinitesimal deformations of Hom-Lie antialgebras. The notion of Nijenhuis operators for Hom-Lie antialgebras is introduced. This kind of operators gives trivial deformation.
Let be a Hom-Lie antialgebra and , be bilinear maps. Consider a -parametrized family of bilinear maps:
[TABLE]
If these maps endow with a Hom-Lie antialgebra structure which is denoted by , then we say that generates a -parameter infinitesimal deformation of Hom-Lie antialgebra .
Theorem 2**.**
With the above notations, generates a -parameter infinitesimal deformation of a Hom-Lie antialgebra if and only if the following two conditions hold:
(i) defines a Hom-Lie antialgebra structure on ;
(ii) is a 2-cocycle of with the coefficients in the adjoint representation.
Proof.
First, if we assume that and are multiplicative Hom-Lie antialgebras, then we have
[TABLE]
the left hand side is equal to
[TABLE]
the right hand side is equal to
[TABLE]
Thus we obtain
[TABLE]
Similarly, one obtain
[TABLE]
Second, assume generates a -parameter infinitesimal deformation of the Hom-Lie antialgebra , then the maps defined above must satisfy (2.8)–(2.11).
For the equality
[TABLE]
the left hand side is equal to
[TABLE]
the right hand side is equal to
[TABLE]
Thus we have
[TABLE]
and
[TABLE]
For the equality
[TABLE]
the left hand side is equal to
[TABLE]
the right hand side is equal to
[TABLE]
Thus we have
[TABLE]
and
[TABLE]
For the equality
[TABLE]
the left hand side is equal to
[TABLE]
the right hand side is equal to
[TABLE]
Thus we have
[TABLE]
and
[TABLE]
For the equality
[TABLE]
we have
[TABLE]
and
[TABLE]
Therefore, by (5.61)–(5.63), (5.65),(5.67),(5.69) and (5.71), defines a multiplicative Hom-Lie antialgebra on . Futhermore, by (5.61)–(5.63), (5.64),(5.66),(5.68) and (5), we obtain that is a 2-cocycle of with coefficients in the adjoint representation.
Conversely, if defines a multiplicative Hom-Lie antialgebra on and it is a 2-cocycle of with coefficients in the adjoint representation, then one can check by the same reasoning as above that generates a -parameter infinitesimal deformation of Hom-Lie antialgebra . This finished the proof. ∎
A deformation is said to be trivial if there exists a linear map such that for , the following holds:
[TABLE]
and
[TABLE]
That is to say is a Hom-Lie antialgebra homomorphism from to .
From equation (5.72), we get
[TABLE]
From equation (5), we have
[TABLE]
and
[TABLE]
Thus, we get
[TABLE]
and
[TABLE]
Similarly,
[TABLE]
[TABLE]
we get
[TABLE]
and
[TABLE]
Next, due to
[TABLE]
and
[TABLE]
we get
[TABLE]
and
[TABLE]
Definition 5.1**.**
A linear operator is called a Nijenhuis operator if and only if(5.74), (5.77), (5.80) and (5.83) hold.
We have seen that every trivial deformation produces a Nijenhuis operator. Conversely, Nijenhuis operator gives a trivial deformation as the following theorem shows.
Theorem 3**.**
Let be a Nijenhuis operator on . Then a deformation of can be obtained by putting
[TABLE]
Furthermore, this deformation is a trivial one.
Proof.
By definition, is a 2-cocycle of with coefficients in the adjoint representation. In the following, we will show that is a Hom-Lie antialgebra of deformation type.
First we check the Hom-associative equality (2.8) hold for . A direct computations shows that
[TABLE]
and
[TABLE]
Since the Hom-associative identity (2.8) hold for , the underline items are cancelled. The remaining items are cancelled by using the Nijenhuis operator condition (5.77) of the forms:
[TABLE]
Thus we get
[TABLE]
Second, by similar computations using the Nijenhuis operator conditions (5.80) and (5.83) we get
[TABLE]
Thus the equality (2.9)–(2.11) hold for . From the above calculations, we obtain that is a Hom-Lie antialgebra. Therefore, satisfies two conditions in Theorem 2 and it gives a trivial deformation. ∎
Acknowledgements
The author would like to thank the referee for careful reading of the manuscript and for valuable suggestions which helped us both in English and in depth to improve the quality of the paper. This research was supported by NSFC(11501179, 11961049) and a doctoral research program of Henan Normal University.
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