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Journal of Lie Theory 16 (2006), No. 2, 271--296
Copyright Heldermann Verlag 2006


P. Lavaud
13 rue Beaunier, 75014 Paris, France


\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.

Keywords: Superanalysis, Lie superalgebras, Lie supergroups, superdeterminant, Berezinian, superpfaffian.

MSC: 17B40, 46F99, 58A50, 58C50

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