Quadratic Lie superalgebras with a reductive even part

Abstract

Communicated by C. Kassel MSC: 17A70 17B05 17B20 a b s t r a c t The aim of this paper is to exhibit some non-trivial examples of quadratic Lie superalgebras such that the even part is a reductive Lie algebra and the action of the even part on the odd part is not completely reducible and to give an inductive description of this class of quadratic Lie superalgebras. The notion of the generalized double extension of quadratic Lie superalgebras proposed by I. Bajo, S. Benayadi and M. Bordemann [I. Bajo, S. Benayadi, M. Bordemann, Generalized double extension and descriptions of quadratic Lie superalgebras, 2007, arXiv:math-ph/0712.0228] is of crucial importance in this work.In particular, we will improve some results of [S. Benayadi, Quadratic Lie superalgebras with the completely reducible action of the even part on the odd part, J. Algebra 223 (2000) 344-366], in the sense that we will not demand that the action of the even part on the odd part is completely reducible, which naturally makes the proofs of our results more difficult.

Quadratic Malcev Superalgebras with Reductive Even Part

“…Analogously to the Lie case (cf. [2]), the socle of M has a remarkable importance in this study and we can prove with the same reasoning that if M is a B-irreducible quadratic Malcev superalgebra with reductive even part that is neither the one dimensional Lie algebra nor a simple Malcev superalgebra the socle coincides with the center. As a consequence of this result, with a proof analogously to the one in Lie case [2], we characterize quadratic Malcev superalgebras with reductive even part that have null center as semisimple Malcev superalgebras.…”

“…In fact we have k = Ke⊕M ⊕Ke * equipped with the even skew-symmetric bilinear map on k defined by We will prove the result by showing the following sequence of claims as it is done in case of the quadratic Lie superalgebras [2], and in case of the odd-quadratic Lie superalgebras [3]. First, we will determine the quadratic Malcev superalgebra (N, B); then we will show that the quadratic Malcev superalgebra (M, B) is the generalized double extension of (N, B) by the one-dimensional Malcev superalgebra (Ke)1.…”

“…The proof is by induction on the dimension of M and is analogous to the corresponding theorem in Lie case (cf. [2]), using the fact that a simple quadratic Malcev superalgebra is either a basic classical Lie superalgebra or is the simple (non Lie) Malcev algebra.…”

On deformations of metric Lie superalgebras

“…In the past years the study E-mail address: yangyong195888221@163.com of metric Lie superalgebras become intensive. Many classes of metric Lie superalgebras have been studied [2,18,19]. Different from what happens in the Lie case, the Killing form is not always non-degenerate on a semi-simple Lie superalgebra.…”

Double extensions of restricted Lie (super)algebras

“…Let a be a Lie superalgebra (in particular, a Lie algebra) equipped with a non-degenerate invariant supersymmetric bilinear form B a , suggestively abbreviated to NIS superalgebra in [BKLS], defined over a field K of positive characteristic p. The notion of a double extension of the Lie superalgebra a, called D-extension in [BeBou], was introduced by Medina and Revoy [MR] in the case of Lie algebras. This notion has been superized and studied in a series of papers [ABB,ABBQ,BBB,BB,B,B2,BeBou]. The double extension of a, denoted by g, simultaneously involves three ingredients:…”