Description de la structure de certaines superalgèbres de Lie quadratiques via la notion de $T^{\ast }$-extension

Abstract

In this note we introduce the notion of $T^{\ast }-$extension $T^{\ast }\mathfrak{g}$ of a Lie superalgebra $\mathfrak{g}$, i.e. an extension of $\mathfrak{g}$ by its dual space ${\mathfrak{g}}^{\ast }$. The natural pairing induces on $T^{\ast }\mathfrak{g}$ an even supersymmetric nondegenerate bilinear form $B$ which is invariant ($B([X,Y],Z)=B(X,[Y,Z])$ for all $X,Y,Z\in T^{\ast }\mathfrak{g}$), 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 $T^{\ast }-$extension in the case of an algebraically closed field of characteristic zero.

LATEX 2e, amssymb, 6 pages, main body of the text in French, abridged English version included