Superpfaffian

Let V = V 0 V 1 be a real finite dimensional supervector space provided with a nondegenerate antisymmetric even bilinear form B .Let spo(V ) be the Lie superalgebra of endomorphisms of V which preserve B .We consider spo(V ) as a supermanifold.We show that a choice of an orientation of V 1 and of a square root i of -1 determines a very interesting generalized function on the supermanifold spo(V ), the superpfaffian.) , the superpfaffian is the usual Pfaffian, a square root of the determinant.When V = V 0 , spo(V ) is the symplectic Lie algebra 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 sp(V 0 ), and it is a square root of the inverse of the determinant in the open subset of invertible elements of spo(V ) .In this article, we present the definition and some basic properties of the superpfaffian.