Quadratic (resp. symmetric) Leibniz superalgebras
Abstract
The aim of this article is to study quadratic Leibniz superalgebras, which are (left or right) Leibniz superalgebras with an even supersymmetric non-degenerate and associative bilinear forms. In particular, we prove that this class of Leibniz superalgebras are symmetric, which are both a left and right Leibniz superalgebras. We give a characterizations of symmetric Leibniz superalgebras. Moreover, we show that every symmetric Leibniz superalgebra is a central extension of a Lie superalgebra by means of an even Leibniz 2-cocycle. We also prove that any solvable quadratic Leibniz superalgebra is a T*-extension of a solvable Lie superalgebra in the category of Leibniz superalgebras. By the procedure of double extension (central extension followed by generalized semi-direct product), we obtain an inductive description of all quadratic Leibniz superalgebras and we reduce the study of this class of Leibniz superalgebras to that of quadratic Lie superalgebras. In particular, this description provides us with an algorithm of construction of quadratic Leibniz superalgebras. Finally, we construct several non-trivial examples of quadratic (resp. symmetric) Leibniz superalgebras.