Odd-quadratic Lie superalgebras

An odd-quadratic Lie superalgebra is a Lie superalgebra with a non-degenerate, supersymmetric, odd, and invariant bilinear form. In this paper we give examples and present some properties of odd-quadratic Lie superalgebras. We introduce the notions of double extension and generalized double extension of odd-quadratic Lie superalgebras and give an inductive description of solvable odd-quadratic Lie superalgebras and of odd-quadratic Lie superalgebras such that the even part is a reductive Lie algebra. We obtain also another interesting description of odd-quadratic Lie superalgebras such that the even part is a reductive Lie algebra without using the notions of double extensions. (C) 2009 Elsevier B.V. All rights reserved.