Quadratic Lie superalgebras with a reductive even part

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.