Description de la structure de certaines superalgèbres de Lie quadratiques via la notion de $T^*$-extension

Abstract

In this note we introduce the notion of $T^*-$extension $T^*{\mathfrak g}$ of a Lie superalgebra ${\mathfrak g}$, i.e. an extension of ${\mathfrak g}$ by its dual space ${\mathfrak g}^*$. The natural pairing induces on $T^*{\mathfrak g}$ an even supersymmetric nondegenerate bilinear form $B$ which is invariant ($B([X,Y],Z)=B(X,[Y,Z])$ for all $X,Y,Z \in T^*{\mathfrak g}$), i.e. the structure of a quadratic (or metrised or orthogonal) Lie superalgebra. These extensions can be classified by the third even scalar cohomology group of ${\mathfrak g}$. Moreover, we show that all finite-dimensional quadratic Lie superalgebras ${\mathfrak a}={\mathfrak a}_{\bar{0}} \oplus {\mathfrak a}_{\bar{1}}$ which are either nilpotent, or solvable and such that $[{\mathfrak a}_{\bar{1}},{\mathfrak a}_{\bar{1}}]\subset [{\mathfrak a}_{\bar{0}},{\mathfrak a}_{\bar{0}}]$ can be constructed by means of a $T^*-$extension in the case of an algebraically closed field of characteristic zero.