Advances in Mathematical Physics

Volume 2014 (2014), Article ID 381683, 9 pages

http://dx.doi.org/10.1155/2014/381683

Research Article

On the Deformations and Derivations of -Ary Multiplicative Hom-Nambu-Lie Superalgebras

Baoling Guan,¹ Liangyun Chen,² and Yao Ma²

¹College of Sciences, Qiqihar University, Qiqihar 161006, China

²School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Received 21 March 2014; Accepted 19 May 2014; Published 5 June 2014

Academic Editor: Yao-Zhong Zhang

Copyright © 2014 Baoling Guan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We introduce the relevant concepts of -ary multiplicative Hom-Nambu-Lie superalgebras and construct three classes of -ary multiplicative Hom-Nambu-Lie superalgebras. As a generalization of the notion of derivations for -ary multiplicative Hom-Nambu-Lie algebras, we discuss the derivations of -ary multiplicative Hom-Nambu-Lie superalgebras. In addition, the theory of one parameter formal deformation of -ary multiplicative Hom-Nambu-Lie superalgebras is developed by choosing a suitable cohomology.

1. Introduction

The notion of -Lie algebras was introduced by Filippov in 1985 in [1]. The -Lie algebra is a vector space endowed with an -ary linear skew-symmetric product which satisfies the generalized Jacobi identity (also named Filippov identity). For this product is a special case of the Nambu bracket, introduced by Nambu in 1973 in [2], and was well known in physics, as a generalization of the Poisson bracket in Hamiltonian mechanics. -Lie algebras are also useful in the research for M2-branes in the string theory and are closely linked to the Plücker relation in the literature in physics in [3–6].

In 1996, the concept of -Lie superalgebras was firstly introduced by Daletskiĭ and Kushnirevich in [7]. Moreover, Cantarini and Kac gave a more general concept of -Lie superalgebras again in 2010 in [8]. -Lie superalgebras are more general structures including -Lie algebras, -ary Nambu-Lie superalgebras, and Lie superalgebras.

The general Hom-algebra structures arose first in connection with quasi-deformation and discretizations of Lie algebras of vector fields. These quasi-deformations lead to quasi-Lie algebras, a generalized Lie algebra structure in which the skew-symmetry and Jacobi conditions are twisted. Hom-Lie algebras, Hom-associative algebras, Hom-Lie superalgebras, Hom-bialgebras, -ary Hom-Nambu-Lie algebras, and quasi-Hom-Lie algebras are discussed in [9–20]. Generalizations of -ary algebras of Lie type and associative type by twisting the identities using linear maps have been introduced in [21].

The mathematical theory of deformations has proved to be a powerful tool in modeling physical reality. For example, (algebras associated with) classical quantum mechanics (and field theory) on a Poisson phase space can be deformed to (algebras associated with) quantum mechanics (and quantum field theory). The deformation of algebraic systems has been one of the problems that many mathematical researchers are interested in; Gerstenhaber studied the deformation theory of algebras in a series of papers [22–26]. For example, it has been extended to covariant functors from a small category to algebras. In [27, 28], it is, respectively, extended to algebra systems, bialgebras, Hopf algebras, Leibniz pairs, Poisson algebras, and so forth. In [23], Gerstenhaber developed a theory of deformation of associative and Lie algebras. His theory links cohomologies of these algebras and the Gerstenhaber bracket giving obstructions to deformations. Nijenhuis and Richardson noticed strong similarities between Gerstenhaber theory and the deformations of complex analytic structures on compact manifolds [29]. They axiomatized the theory of deformations via the introduction of graded Lie algebras [30]. One such example was given by the theory of deformations of homomorphisms [31]. Inspired by these works, we study the deformation theory of -ary multiplicative Hom-Nambu-Lie superalgebras in this paper. In addition, the paper also discusses derivations of -ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notions of derivations for -ary multiplicative Hom-Nambu-Lie algebras.

This paper is organized as follows. In Section 1, we introduce the relevant concepts of -ary multiplicative Hom-Nambu-Lie superalgebras and construct three classes of -ary multiplicative Hom-Nambu-Lie superalgebras. In Section 2, the notion of derivation introduced for -ary multiplicative Hom-Nambu-Lie algebras in [10] is extended to -ary multiplicative Hom-Nambu-Lie superalgebras. In Section 3, the theory of deformations of -ary multiplicative Hom-Nambu-Lie superalgebras is developed by choosing a suitable cohomology.

Definition 1 (see [32]). An -ary Nambu-Lie superalgebra is a pair consisting of a -graded vector space and a multilinear mapping , satisfying

where denotes the degree of a homogeneous element .

Definition 2. 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 with and satisfying

If the -ary Hom-Nambu-Lie superalgebra is multiplicative, then (4) can be read as

It is clear that -ary Hom-Nambu-Lie algebras and Hom-Lie superalgebras are particular cases of -ary Hom-Nambu-Lie superalgebras. In the sequel, when the notation “” appears, it means that is a homogeneous element of degree .

Definition 3. Let and be two -ary Hom-Nambu-Lie superalgebras, where and . A linear map is an -ary Hom-Nambu-Lie superalgebra morphism if satisfies

Theorem 4. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra and let be a morphism of such that . Then is an -ary multiplicative Hom-Nambu-Lie superalgebra.

Proof. Put . Then

that is, is a morphism of . Moreover, we have

Therefore, is an -ary multiplicative Hom-Nambu-Lie superalgebra.

In particular, we have the following example.

Example 5. 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 6. Let be an -ary Hom-Nambu-Lie superalgebra. A graded subspace is a Hom-subalgebra of if and is closed under the bracket operation ,; that is, , .

A graded subspace is a Hom-ideal of if and , , .

Definition 7. Let and be two -ary multiplicative Hom-Nambu-Lie superalgebras. Suppose that is a linear map. is called as the graph of a linear map .

Proposition 8. Given two -ary multiplicative Hom-Nambu-Lie superalgebras and , there is an -ary multiplicative Hom-Nambu-Lie superalgebra , where the bilinear map is given by

and the linear map is given by

Proof. For any , we have

The bracket is obviously supersymmetric. By a direct computation we have

The result follows.

Proposition 9. A linear map is a morphism of -ary multiplicative Hom-Nambu-Lie superalgebras if and only if the graph is a Hom-subalgebra of .

Proof. Let be a morphism of -ary multiplicative Hom-Nambu-Lie superalgebras. Then

Then the graph is closed under the bracket operation . Furthermore, we obtain

which implies that . Thus, is a Hom-subalgebra of .

Conversely, if the graph is a Hom-subalgebra of , then we have

which implies that

Furthermore, yields that

which is equivalent to the condition ; that is, . Therefore, is a morphism of -ary multiplicative Hom-Nambu-Lie superalgebras.

2. Derivations of -Ary Multiplicative Hom-Nambu-Lie Superalgebras

Let be an -ary multiplicative Hom-Nambu-Lie superalgebra. We denote by the -times compositions of . In particular, we set .

Definition 10. For , we call an -derivation of the -ary multiplicative Hom-Nambu-Lie superalgebra if

and for ,

We denote by the set of -derivations of the -ary multiplicative Hom-Nambu-Lie superalgebra . Notice that we obtain classical derivations for .

For satisfying and , we define the map by

Then one has the following.

Lemma 11. The map is an -derivation and is called an inner -derivation.

We denote by the -vector space generated by all inner -derivations. For any and , we define their commutator . Set and .

Lemma 12. For any and , one has .

Proof. Let , . and , and then

Then we obtain

which yields that .

Proposition 13. The pair , where the bracket is the usual commutator, defines a Lie superalgebra and constitutes an ideal of it.

Proof. is a Lie superalgebra by using Lemma 12. We show that is an ideal. Let be an inner -derivation on and for and . Then and for any

Therefore, .

3. Deformations of -Ary Multiplicative Hom-Nambu-Lie Superalgebras

Definition 14 (see [33]). For , we call -coboundary operator of the -ary multiplicative Hom-Nambu-Lie superalgebra the even linear map by

where , , and the last term is defined by

Theorem 15 (see [33]). Let be an -cochain. Then .

In [33], it also points out that 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 factor space .

We next will discuss the deformation of -ary multiplicative Hom-Nambu-Lie superalgebras. Let denote the power series ring in one variable with coefficients in and let be the set of formal series whose coefficients are elements of the vector space .

Definition 16. Let be an -ary multiplicative Hom-Nambu-Lie superalgebra over . A deformation of is given by --linear map

such that is also an -ary multiplicative Hom-Nambu-Lie superalgebra. We call the infinitesimal deformation of .

Since is an -ary multiplicative Hom-Nambu-Lie superalgebra, satisfies

Equations (27)–(29) are, respectively, equivalent to

We call these the deformation equations for an -ary multiplicative Hom-Nambu-Lie superalgebra.

Equations and show that . In , set , and then

that is, . Hence the infinitesimal deformation .

Definition 17. Two deformations and of the -ary multiplicative Hom-Nambu-Lie superalgebra are said to be equivalent, if there exists an isomorphism of -ary multiplicative Hom-Nambu-Lie superalgebras , where , is a linear map such that

and is denoted by . When is called the null deformation; if , then is called the trivial deformation.

Theorem 18. Let and be two equivalent deformations of the -ary multiplicative Hom-Nambu-Lie superalgebra . Then the infinitesimal deformations and belong to the same cohomology class in the cohomology group .

Proof. Put . It is enough to prove that . Let be an isomorphism of -ary multiplicative Hom-Nambu-Lie superalgebras. Then and

and comparing with the coefficients of for two sides of the above equation, we obtain

Furthermore, one gets

Therefore, ; that is, .

An -ary multiplicative Hom-Nambu-Lie superalgebra is analytically rigid if every deformation is equivalent to the null deformation . We have a fundamental theorem.

Theorem 19. If is an -ary multiplicative Hom-Nambu-Lie superalgebra with , then is analytically rigid.

Proof. Let be a deformation of the -ary multiplicative Hom-Nambu-Lie superalgebra with ; that is; . Then setting in , we have

that is, , ; that is, . By our assumption , one gets , and thus we can find such that . Putting , then

and moreover, . Hence is a linear isomorphism and . Set , and then is also a deformation of and . Note that . Let . Then

So

By the above equation, one gets

Furthermore, we have

and hence, . By induction, one can prove ; that is, is analytically rigid.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by NNSF of China (no. 11171055), NSF of Jilin (no. 201115006), Scientific Research Fund of Heilongjiang Provincial Education Department (no. 12541900), Scientific Research Foundation for Returned Scholars Ministry of Education of China, and the Fundamental Research Funds for the Central Universities (no. 12SSXT139).

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