On the Cohomology and Extensions of -ary Multiplicative Hom-Nambu-Lie Superalgebras

Abstract

In this paper, we discuss the representations of -ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notion of representations for -ary multiplicative Hom-Nambu-Lie algebras. We also give the cohomology of an -ary multiplicative Hom-Nambu-Lie superalgebra and obtain a relation between extensions of an -ary multiplicative Hom-Nambu-Lie superalgebra by an abelian one and . We also introduce the notion of -extensions of -ary multiplicative Hom-Nambu-Lie superalgebras and prove that every finite-dimensional nilpotent metric -ary multiplicative Hom-Nambu-Lie superalgebra over an algebraically closed field of characteristic not 2 in the case is a surjection is isometric to a suitable -extension.

1. Introduction

In 1996, the concept of -Lie superalgebras was firstly introduced by Daletskii and Kushnirevich in [1]. Moreover, Cantarini and Kac gave a more general concept of -Lie superalgebras again in 2010 in [2]. -Lie superalgebras are more general structures including -Lie algebras (-ary Nambu-Lie algebras), -ary Nambu-Lie superalgebras, and Lie superalgebras [3].

The general Hom-algebra structures arose first in connection with quasideformation and discretizations of Lie algebras of vector fields. These quasideformations lead to quasi-Lie algebras, a generalized Lie algebra structure in which the skew symmetry and Jacobi conditions are twisted. Hom-Lie algebras, Hom-Lie superalgebras, Hom-Lie bialgebras, Hom-Lie -algebras, and quasi-Hom-Lie algebras are discussed in [4–14]. The -ary Hom-Nambu-Lie algebras have been introduced in [15]. It is the generalization of -ary algebras of Lie type by twisting the identities using linear maps. It includes -ary Hom-algebra structures generalizing the -ary algebras of Lie type such as -ary Nambu algebras, -ary Nambu-Lie algebras, and -ary Lie algebras [16].

Cohomologies are powerful tools in mathematics, which can be applied to algebras and topologies as well as the theory of smooth manifolds or of holomorphic functions. The cohomology of Lie algebras was defined by Chevalley and Eilenberg in order to give an algebraic construction of the cohomology of the underlying topological spaces of compact Lie groups in [17]. The cohomology of Lie superalgebras was introduced by Scheunert and Zhang in [18] and was used in mathematics and theoretical physics: the theory of cobordisms, invariant differential operators, central extensions, and deformations. The theory of cohomology for -ary Hom-Nambu-Lie algebras and -Lie superalgebras can be found in [19, 20]. This paper generalizes it to -ary multiplicative Hom-Nambu-Lie superalgebras.

The extension is an important way to find a larger algebra and there are many extensions such as general extensions, abelian extensions, nonabelian extensions, double extensions, and Kac-Moody extensions. Abelian extensions and nonabelian extensions of Hom-Lie algebras are, respectively, researched in [21, 22]. The general extensions of -Hom-Lie algebras is researched in [23]. In 1997, Bordemann introduced the notion of -extensions of Lie algebras in [24]. The method of -extension was used in [25] and was generalized to many other algebras recently in [26, 27]. This paper researches general extensions and -extensions of -ary multiplicative Hom-Nambu-Lie superalgebras. In addition, the paper also discusses representations of -ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notions of representations for -ary multiplicative Hom-Nambu-Lie algebras.

This paper is organized as follows. In Section 2, we give the representation and the cohomology for an -ary multiplicative Hom-Nambu-Lie superalgebra. In Section 3, we give a one-to-one correspondence between extensions of an -ary multiplicative Hom-Nambu-Lie superalgebras by an abelian one and In Section 4, we introduce the notion of -extensions of -ary multiplicative Hom-Nambu-Lie superalgebras and prove that every finite-dimensional nilpotent metric -ary multiplicative Hom-Nambu-Lie superalgebra over an algebraically closed field of characteristic not 2 such that is isometric to (a nondegenerate ideal of codimension of) a -extension of a nilpotent -ary multiplicative Hom-Nambu-Lie superalgebra whose nilpotent length is at most a half of the nilpotent length of .

2. -ary Hom-Nambu-Lie Superalgebras

In the paper, let be a finite-dimensional -graded vector space. The degree of an element in will be denoted by and in what follows appearance of will mean that is a homogeneous element and stands for its degree, where and

Definition 1. An -ary Hom-Nambu-Lie superalgebra is a triple consisting of a -graded vector space a multilinear mapping and a family of even linear maps satisfying where denotes the degree of a homogeneous element

An -ary Hom-Nambu-Lie superalgebra is multiplicative, if and the following equality is satisfied:

Moreover, the multiplicative -ary Hom-Nambu-Lie superalgebra is also denoted by

For a multiplicative -ary Hom-Nambu-Lie superalgebra equation (4) can be read:

It is clear that -ary Hom-Nambu-Lie algebras and Hom-Lie superalgebras are particular cases of -ary Hom-Nambu-Lie superalgebras.

Definition 2. Let and be two -ary Hom-Nambu-Lie superalgebras. A linear map is an -ary Hom-Nambu-Lie superalgebra homomorphism if it satisfies

Example 3. Let be an -ary Nambu-Lie superalgebra and let be an -ary Nambu-Lie superalgebra endomorphism. Then, is an -ary multiplicative Hom-Nambu-Lie superalgebra.

Definition 4. Let be an -ary Hom-Nambu-Lie superalgebra. A graded subspace is a Hom-subalgebra of if and is closed under the bracket operation i.e., for all A graded subspace is a Hom-ideal of if and for all and

Definition 5. Let be an -ary Hom-Nambu-Lie superalgebra. A Hom-ideal of is abelian if for all and

3. Cohomology for -ary Multiplicative Hom-Nambu-Lie Superalgebras

Definition 6. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra. is called a fundamental object of For all . It is clear that .
Let and be two fundamental objects of . A bilinear map defined by A linear map defined by Then, .

Proposition 7. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra. Suppose that , and are fundamental objects of and is an arbitrary element in . Then,

Proof. It is easy to see that (9) is equivalent to (6). Using (9), by exchanging and , we have Comparing (9) with (11), we obtain (10).

Definition 8. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra and be a -graded vector space over a field and A graded representation of on is a linear map such that for and where the sign indicates that the element below must be omitted. The -graded representation space is said to be a graded -module.

We use a supersymmetric notation (like (3)) to denote Set and for all and then becomes an -ary multiplicative Hom-Nambu-Lie superalgebra such that is a -graded abelian ideal of , that is,

In the sequel, we will usually abbreviate with .

Example 9. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra. Then, defined by is a graded representation of it is also called the adjoint graded representation of

Definition 10. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra and be a graded -module. An -cochain is an -linear map such that for all and We denote the set of -cochain by

Definition 11. For we define an -coboundary operator of the -ary multiplicative Hom-Nambu-Lie superalgebra which is an even linear map, by where and the last term is defined by where

Theorem 12. Let be an -cochain. Then,

Proof. See the appendix.

Remark 13. The -coboundary operator as above is a generalization of the one defined for -ary multiplicative Hom-Nambu-Lie algebras in [16] and for first-class -Lie superalgebras in [20].

The map is called an -supercocycle if . We denote by the graded subspace spanned by -supercocycles. Since for all , is a graded subspace of . Therefore, we can define a graded cohomology space of as the graded space

4. Extensions of -ary Multiplicative Hom-Nambu-Lie Superalgebras

Definition 14. Let be a family of -ary multiplicative Hom-Nambu-Lie superalgebras over . is a morphism of -ary multiplicative Hom-Nambu-Lie superalgebras. The sequence is called an exact sequence of -ary multiplicative Hom-Nambu-Lie superalgebras, if it satisfies

Definition 15. Let and be -ary multiplicative Hom-Nambu-Lie superalgebras over . is called an extension of by if there is an exact sequence of -ary multiplicative Hom-Nambu-Lie superalgebras:

Let and be two -ary multiplicative Hom-Nambu-Lie superalgebras over . Suppose that is an abelian graded ideal of , i.e., is a graded ideal such that We consider the case that is an extension of by an abelian graded ideal of . Let be a homogeneous even linear map with and Let and let . Then, becomes a graded -module. Let us write and then denote the elements of by for all and . Then, the bracket in is defined by where and . It is easy to see that . Let and . Then,

Therefore, .

Conversely, suppose that an abelian -ary multiplicative Hom-Nambu-Lie superalgebras is a graded -module, , and . Let where Then is an -ary multiplicative Hom-Nambu-Lie superalgebra with the bracket defined by (24). Then we can define an exact sequence where . Thus is an extension of by and is an abelian graded ideal of .

Therefore, we get the following theorem.

Theorem 16. Suppose that and are two -ary multiplicative Hom-Nambu-Lie superalgebras over and is abelian. Then, there is a one-to-one correspondence between extensions of by and .

5. -Extensions of -ary Multiplicative Hom-Nambu-Lie Superalgebras

Let be an -ary multiplicative Hom-Nambu-Lie superalgebra and be its dual space. Since and are -graded vector space, the direct sum is a -graded vector space. In the sequel, whenever appears, it means that is homogeneous and .

Lemma 17. Let be the dual -graded vector space of an -ary multiplicative Hom-Nambu-Lie superalgebra . Let us consider the even linear map defined by for all and . Then, is a representation of on if and only if the following conditions hold: for all

Proof. We firstly prove that the necessity holds. Then, by the definition of one gets and Moreover, we have By (13), we have By (14), we obtain It is easy to see that the sufficiency holds. The proof is complete.

The representation as defined in Lemma 17 is called the coadjoint representation of Let be a homogeneous -linear map from into of degree 0. Now, we define a bracket on :

Theorem 18. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra. Assume that the coadjoint representation exists. Then, is an -ary multiplicative Hom-Nambu-Lie superalgebra if and only if where