The general linear supergroup and its Hopf superalgebra of regular functions

Abstract

The structure of the Hopf superalgebra B of regular functions on the general linear supergroup is developed, and applied to study the representation theory of the supergroup. It is shown that the general linear supergroup can be reconstructed from B in a way reminiscent of the Tannaka–Krein theory. A Borel–Weil type realization is also obtained for the irreducible integrable representations. As a side result on the structure of B , Schur superalgebras are introduced and are shown to be semi-simple over the complex field, with the simple ideals determined explicitly.