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 Z 2 -graded structures giving a notion of Lie superalgebra. Analogously a notion of n-Lie algebra can be extended to Z 2 -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 Z 2 -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.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
We construct 3-Lie superalgebras on a commutative superalgebra by means of involution and even degree derivation. We construct a representation of induced 3-Lie algebras and superalgebras by means of a representation of initial (binary) Lie algebra, trace and supertrace. We show that the induced representation of 3-Lie algebra, that we constructed, is a representation by traceless matrices, that is, lies in the Lie algebra sl(V ), where V is a representation space. In the case of 2-dimensional representation we find conditions under which the induced representation of induced 3-Lie algebra is irreducible. We give the example of irreducible representation of induced 3-Lie algebra of 2nd order complex matrices.
This volume contains a selection of papers written on the basis of presentations given at the 10th International Conference on Clifford Algebras and their Applications in Mathematical Physics (ICCA10), August 4-9, 2014, in Tartu, Estonia. The conference presentations can be grouped into four sessions in Clifford analysis, Clifford algebras, geometry and physics, and, according to this classification, the volume consists of four parts which have the same titles.This conference was a continuation of tradition to bring together the leading scientists in the field of Clifford analysis, Clifford algebras, geometry and their applications in various fields of theoretical physics by organizing scientific meetings. Previous conferences on Clifford algebras and their applications were held at
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