Capelli's theory, Koszul maps, and superalgebras

The straightening laws for the enveloping algebra Ukappa(pl(L)) of the general linear Lie superalgebra of a finite dimensional Z2-graded vector space are investigated. An isomorphism Psi from the supersymmetric algebra Super[L L] of pl(L) to Ukappa(pl(L)) is introduced; the isomorphism Psi maps each bitableau of Super[L L] to the Young-Capelli bitableau of Ukappa(pl(L)) parametrized by the same pair of Young diagrams, both in the permanental case and in the determinantal case. The map Psi is shown to be the inverse of the isomorphism introduced by Koszul [Koszul, J. P. (1981) C.R. Acad. Sci. Paris 292, 139-141]. The set of all costandard determinantal Young-Capelli bitableaux is a basis of Ukappa(pl(L)); this basis acts in a triangular way on the basis of Super[L L]given by the set of all standard permanental bitableaux.