Super 3-Lie Algebras Induced by Super Lie Algebras

Abramov, Viktor

doi:10.1007/s00006-015-0604-3

Cited by 32 publications

(36 citation statements)

References 5 publications

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“…24, 2018 22:17 WSPC/INSTRUCTION FILE matrix˙3-Lie˙superalgebras˙BRST 10 We see that the term (33) at the left-hand side of the identity is canceled by two terms (35) of the right-hand side of identity because all together they form the usual graded Jacobi identity. Analogously it can be shown that all the rest five terms at the left-hand side of the identity (32) are canceled by the corresponding terms of the right-hand side of the identity.…”

Section: Now We Define a Graded Triple Lie Bracket Of Matricesmentioning

confidence: 89%

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Matrix 3-Lie superalgebras and BRST supersymmetry

Abramov

2017

Int. J. Geom. Methods Mod. Phys.

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Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper we show that this approach can be extended to the infinitedimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras gl (m, n) if instead of the trace of a matrix we make use of the super trace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the super trace satisfies a graded ternary Filippov-Jacobi identity. In two particular cases of gl (1, 2) and gl(2, 2) we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

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Section: Now We Define a Graded Triple Lie Bracket Of Matricesmentioning

confidence: 89%

“…In the case of a matrix Lie superalgebra glt(m, n) we define a graded triple commutator of three matrices by a formula similar to (1), where instead of the trace of a matrix we use the super trace and graded binary commutator [10], [11]. We prove that a graded triple commutator of matrices satisfies the graded Filippov-Jacobi identity.…”

Section: Introductionmentioning

confidence: 99%

Matrix 3-Lie superalgebras and BRST supersymmetry

Abramov

2017

Int. J. Geom. Methods Mod. Phys.

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“…In [1], let V = V0 ⊕ V1 be a finite-dimensional Z 2 -graded vector space. If v ∈ V is a homogenous element, then its degree will be denoted by |v|, where |v| ∈ Z 2 and Z 2 = {0,1}.…”

Section: -Ary Lie Superalgebrasmentioning

confidence: 99%

3-ary Hom-Lie Superalgebras Induced By Hom-Lie Superalgebras

Guan

Chen

Sun

2017

Adv. Appl. Clifford Algebras

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The purpose of this paper is to study the relationships between a Hom-Lie superalgebra and its induced 3-ary-Hom-Lie superalgebra. We provide an overview of the theory and explore the structure properties such as ideals, center, derived series, solvability, nilpotency, central extensions, and the cohomology.1 of how to construct ternary multiplications from the binary multiplication of a Hom-Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions; and it was shown that this method can be used to construct ternary Hom-Nambu-Lie algebras from Hom-Lie algebras. This construction was generalized to n-Lie algebras and n-Hom-Nambu-Lie algebras in [5].The reference [1] constructed super 3-Lie algebras(that is, 3-ary Lie superalgebras) by super Lie algebras(that is, Lie superalgebras). The reference [14] constructed (n + 1)-Hom-Lie algebras by n-Hom-Lie algebras. Inspired by the references [14] and [1], the paper generalizes them to the case of Hom-superalgebra. Its aim is to study the relationships between a Hom-Lie superalgebra and its induced 3-ary-Hom-Lie superalgebra. We explore the structure properties of objects such as ideals, center, derived series, solvability, nilpotency, central extensions, and the cohomology. In Section 2, we recall the basic notions for Hom-Lie superalgebras and the construction of 3-ary-Hom-Lie superalgebras induced by Hom-Lie superalgebras, we also give a new construction theorem for such algebras and give some basic properties. In Section 3, we define the notion of solvability and nilpotency for 3-ary-Hom-Lie superalgebras and discuss solvability and nilpotency of 3-ary-Hom-Lie superalgebras induced by Hom-Lie superalgebras. In Section 4, we recall the definition and main properties of central extensions of Hom-Lie superalgebras, and then we study central extensions of 3-ary-Hom-Lie superalgebras induced by Hom-Lie superalgebras. The Section 5 is dedicated to study the corresponding cohomology.

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“…It is well known that a concept of binary Lie algebra can be extended to Z 2 -graded structures giving a notion of Lie superalgebra. Similarly a notion of n-Lie algebra can be extended to Z 2 -graded structures giving a structure which we call an n-Lie superalgebra [2,4]. In this section we give the definitions of n-Lie algebra, n-Lie superalgebra and show that a structure of induced n-Lie algebra based on an analog of trace [3] can be extended to n-Lie superalgebras with the help of supertrace.…”

Section: Supertrace and Induced N-lie Superalgebrasmentioning

confidence: 99%

Classification of Low Dimensional 3-Lie Superalgebras

Abramov

Lätt

2016

Springer Proceedings in Mathematics &Amp; Statistics

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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

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Super 3-Lie Algebras Induced by Super Lie Algebras

Cited by 32 publications

References 5 publications

Matrix 3-Lie superalgebras and BRST supersymmetry

Matrix 3-Lie superalgebras and BRST supersymmetry

3-ary Hom-Lie Superalgebras Induced By Hom-Lie Superalgebras

Classification of Low Dimensional 3-Lie Superalgebras

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