Classification of Low Dimensional 3-Lie Superalgebras
A notion of n-Lie algebra introduced by V.T. Filippov can be viewed as a generalization of a concept of binary Lie algebra to the algebras with n-ary multiplication law. A notion of Lie algebra can be extended to \(\mathbb Z_2\) -graded structures giving
- PDF / 169,583 Bytes
- 12 Pages / 439.37 x 666.142 pts Page_size
- 25 Downloads / 232 Views
Abstract A notion of n-Lie algebra introduced by V.T. Filippov can be viewed as a generalization of a concept of binary Lie algebra to the algebras with n-ary multiplication law. A notion of Lie algebra can be extended to Z2 -graded structures giving a notion of Lie superalgebra. Analogously a notion of n-Lie algebra can be extended to Z2 -graded structures by means of a graded Filippov identity giving a notion of n-Lie superalgebra. We propose a classification of low dimensional 3-Lie superalgebras. We show that given an n-Lie superalgebra equipped with a supertrace one can construct the (n + 1)-Lie superalgebra which is referred to as the induced (n + 1)-Lie superalgebra. A Clifford algebra endowed with a Z2 -graded structure and a graded commutator can be viewed as the Lie superalgebra. It is well known that this Lie superalgebra has a matrix representation which allows to introduce a supertrace. We apply the method of induced Lie superalgebras to a Clifford algebra to construct the 3-Lie superalgebras and give their explicit description by ternary commutators. Keywords n-Lie algebras · n-Lie superalgebras · Clifford algebras · Induced n-Lie superalgebras
1 Introduction Recently, there was markedly increased interest of theoretical physics towards the algebras with n-ary multiplication law. Due to the fact that the Lie algebras play a crucial role in theoretical physics, it seems that development of n-ary analog of a concept of Lie algebra is especially important. In [5] V.T. Filippov proposed a notion of n-Lie algebra which can be considered as a possible generalization of a concept V. Abramov (B) · P. Lätt Institute of Mathematics and Statistics, University of Tartu, Liivi 2–602, 50409 Tartu, Estonia e-mail: [email protected] P. Lätt e-mail: [email protected] © Springer International Publishing Switzerland 2016 S. Silvestrov and M. Ranˇci´c (eds.), Engineering Mathematics II, Springer Proceedings in Mathematics & Statistics 179, DOI 10.1007/978-3-319-42105-6_1
1
2
V. Abramov and P. Lätt
of Lie algebra to structures with n-ary multiplication law. In approach proposed by V.T. Filippov an n-ary commutator of n-Lie algebra is skew-symmetric and satisfies an n-ary analog of Jacobi identity which is now called Filippov identity. It is worth to mention that there is an approach different from the one proposed by V.T. Filippov, where a ternary commutator is not skew-symmetric but it obeys a symmetry based on a representation of the group of cyclic permutations Z3 by cubic roots of unity [2]. It is well known that a concept of Lie algebra can be extended to Z2 -graded structures with the help of graded commutator and graded Jacoby identity, and a corresponding structure is known under the name of Lie superalgebra. In the present paper we show that a notion of n-Lie algebra proposed by V.T. Filippov can be extended to Z2 -graded structures by means of graded n-commutator and a graded analog of Filippov identity. This Z2 -graded n-Lie algebra will be referred to as a n-Lie superalgebra. We show that a method of indu
Data Loading...
