Advances in Mathematical Physics

Volume 2014, Article ID 381683, 9 pages

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

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

^{1}College of Sciences, Qiqihar University, Qiqihar 161006, China^{2}School 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).

#### References

- V. T. Filippov, “$n$-Lie algebras,”
*Akademiya Nauk SSSR. Sibirskoe Otdelenie. Sibirskiĭ Matematicheskiĭ Zhurnal*, vol. 26, no. 6, pp. 126–140, 1985 (Russian). View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Nambu, “Generalized Hamiltonian dynamics,”
*Physical Review D*, vol. 7, pp. 2405–2412, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Bagger and N. Lambert, “Gauge symmetry and supersymmetry of multiple M2-branes,”
*Physical Review D*, vol. 77, no. 6, Article ID 065008, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - J. Bagger and N. Lambert, “Comments on multiple M2-branes,”
*Journal of High Energy Physics*, vol. 105, no. 2, p. 15, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - A. Gustavsson, “Algebraic structures on parallel M2 branes,”
*Nuclear Physics B*, vol. 811, no. 1-2, pp. 66–76, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Papadopoulos, “M2-branes, 3-Lie algebras and Plücker relations,”
*Journal of High Energy Physics*, no. 5, p. 9, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Yu. L. Daletskiĭ and V. A. Kushnirevich, “Inclusion of the Nambu-Takhtajan algebra in the structure of formal differential geometry,”
*Dopovidi Natsional'noyi Akademiyi Nauk Ukrayini*, no. 4, pp. 12–18, 1996. View at Google Scholar · View at MathSciNet - N. Cantarini and V. G. Kac, “Classification of simple linearly compact $n$-Lie superalgebras,”
*Communications in Mathematical Physics*, vol. 298, no. 3, pp. 833–853, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - F. Ammar, Z. Ejbehi, and A. Makhlouf, “Cohomology and deformations of Hom-algebras,”
*Journal of Lie Theory*, vol. 21, no. 4, pp. 813–836, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Ammar, S. Mabrouk, and A. Makhlouf, “Representations and cohomology of $n$-ary multiplicative Hom-Nambu-Lie algebras,”
*Journal of Geometry and Physics*, vol. 61, no. 10, pp. 1898–1913, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - F. Ammar and A. Makhlouf, “Hom-Lie superalgebras and Hom-Lie admissible superalgebras,”
*Journal of Algebra*, vol. 324, no. 7, pp. 1513–1528, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - R. P. Bai and Y. Li, “Extensions of $n$-Hom Lie algebras,”
*Frontiers of Mathematics in China*, 2014. View at Publisher · View at Google Scholar - S. Benayadi and A. Makhlouf, “Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms,”
*Journal of Geometry and Physics*, vol. 76, pp. 38–60, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - J. T. Hartwig, D. Larsson, and S. D. Silvestrov, “Deformations of Lie algebras using $\sigma $-derivations,”
*Journal of Algebra*, vol. 295, no. 2, pp. 314–361, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - D. Larsson and S. D. Silvestrov, “Quasi-hom-Lie algebras, central extensions and 2-cocycle-like identities,”
*Journal of Algebra*, vol. 288, no. 2, pp. 321–344, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - A. Makhlouf and S. Silvestrov, “Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras,”
*Forum Mathematicum*, vol. 22, no. 4, pp. 715–739, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - A. Makhlouf and S. Silvestrov, “Hom-algebras and Hom-coalgebras,”
*Journal of Algebra and its Applications*, vol. 9, no. 4, pp. 553–589, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Sheng, “Representations of hom-Lie algebras,”
*Algebras and Representation Theory*, vol. 15, no. 6, pp. 1081–1098, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Yau, “The Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras,”
*Journal of Physics A. Mathematical and Theoretical*, vol. 42, no. 16, Article ID 165202, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Yau, “Hom-algebras and homology,”
*Journal of Lie Theory*, vol. 19, no. 2, pp. 409–421, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Ataguema, A. Makhlouf, and S. Silvestrov, “Generalization of $n$-ary Nambu algebras and beyond,”
*Journal of Mathematical Physics*, vol. 50, no. 8, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - M. Gerstenhaber, “The cohomology structure of an associative ring,”
*Annals of Mathematics*, vol. 78, pp. 267–288, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Gerstenhaber, “On the deformation of rings and algebras,”
*Annals of Mathematics*, vol. 79, pp. 59–103, 1964. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Gerstenhaber, “On the deformation of rings and algebras. II,”
*Annals of Mathematics*, vol. 84, pp. 1–19, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Gerstenhaber, “On the deformation of rings and algebras. III,”
*Annals of Mathematics*, vol. 88, pp. 1–34, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Gerstenhaber, “On the deformation of rings and algebras. IV,”
*Annals of Mathematics*, vol. 99, pp. 257–276, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Gerstenhaber and S. D. Schack, “Algebras, bialgebras, quantum groups, and algebraic deformations,” in
*Deformation Theory and Quantum Groups with Applications to Mathematical Physics*, M. Gerstenhaber and J. Stasheff, Eds., vol. 134 of*Contemporary Mathematics*, pp. 51–92, American Mathematical Society, Providence, RI, USA, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Flato, M. Gerstenhaber, and A. A. Voronov, “Cohomology and deformation of Leibniz pairs,”
*Letters in Mathematical Physics*, vol. 34, no. 1, pp. 77–90, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Nijenhuis and R. W. Richardson, Jr., “Cohomology and deformations of algebraic structures,”
*Bulletin of the American Mathematical Society*, vol. 70, pp. 406–411, 1964. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Nijenhuis and R. W. Richardson Jr., “Cohomology and deformations in graded Lie algebras,”
*Bulletin of the American Mathematical Society*, vol. 72, pp. 1–29, 1966. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Nijenhuis and R. W. Richardson Jr., “Deformations of homomorphisms of Lie groups and Lie algebras,”
*Bulletin of the American Mathematical Society*, vol. 73, pp. 175–179, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Ammar and N. Saadaoui, “Cohomology of $n$-ary-Nambu-Lie superalgebras and super ${\omega}_{\infty}$3-algebra,” http://arxiv-web3.library.cornell.edu/abs/1304.5767v1.
- B. L. Guan, L. Y. Chen, and Y. Ma, “On the cohomology and extensions of $n$-ary multiplicative Hom-Nambu-Lie superalgebras,” http://arxiv.org/abs/1401.0378.