The Centroid And Invariant Bilinear Forms On Lie Superalgebras

Throughout the present paper, we mainly discuss the properties of the centroid of finite dimensional Lie super algebras and quadratic Lie superalgebras.In the paper [1], V. G. Kac have given the concept of the centroid of superalgebras. Through it, we obtain the definition of the centroid of Lie superalgebras. At the same time, we also give the definitions of superfiled and of zero degree centroid of Lie superalgebras. Then we investigate the properties of the centroid and bilinear forms of simple Lie superalgebras. It is well known that the Killing form of every simple Lie superalgebra is either non-degenerate or identically zero. Summarizing the character of non-degenerate Killing form case, we investigate the structure of Lie superalgebras with a class of special bilinear forms called invariant scalar products satisfying consistency, supersymmetry, non-degeneracy and invariance. In other words, we study the properties of quadratic Lie superalgebras. In the last part of this paper our work is devoted to study real Lie superalgebras.