Superpfaffian

P. Lavaud

Journal of Lie theory, Tome 16 (2006) no. 2, pp. 271-296

Résumé

\def\0{{\bf0}}\def\1{{\bf1}}\def\ii{{\bf i}} Let $V=V_\0\oplus V_\1$ be a real finite dimensional supervector space provided with a nondegenerate antisymmetric even bilinear form $B$. Let $\frak{spo}(V)$ be the Lie superalgebra of endomorphisms of $V$ which preserve $B$. We consider $\frak{spo}(V)$ as a supermanifold. We show that a choice of an orientation of $V_\1$ and of a square root $\ii$ of $-1$ determines a very interesting generalized function on the supermanifold $\frak{spo}(V)$, the {\it superpfaffian}. When $V=V_\1$, $\frak{spo}(V)$ is the orthogonal Lie algebra $\frak{so}(V_\1)$, the superpfaffian is the usual Pfaffian, a square root of the determinant. When $V=V_\0$, $\frak{spo}(V)$ is the symplectic Lie algebra $\frak{sp}(V_\0)$, the superpfaffian is a constant multiple of the Fourier transform of one the two minimal nilpotent orbits in the dual of the Lie algebra $\frak{sp}(V_\0)$, and it is a square root of the inverse of the determinant in the open subset of invertible elements of $\frak{spo}(V)$. In this article, we present the definition and some basic properties of the superpfaffian.

Détail

Classification : 17B40, 46F99, 58A50, 58C50
Mots-clés : Superanalysis, Lie superalgebras, Lie supergroups, superdeterminant, Berezinian, superpfaffian