QUANTUM SUPERGROUPS OF GL(n|m) TYPE: DIFFERENTIAL FORMS, KOSZUL COMPLEXES AND BEREZINIANS VOLODIMIR LYUBASHENKO AND ANTHONY SUDBERY

We introduce and study the Koszul complex for a Hecke R-matrix. Its cohomologies, called the Berezinian, are used to define quantum superdeterminant for a Hecke R-matrix. Their behaviour with respect to Hecke sum of R-matrices is studied. Given a Hecke R-matrix in n-dimensional vector space, we construct a Hecke R-matrix in 2n-dimensional vector space commuting with a differential. The notion of a quantum differential supergroup is derived. Its algebra of functions is a differential coquasitriangular Hopf algebra, having the usual algebra of differential forms as a quotient. Examples of superdeterminants related to these algebras are calculated. Several remarks about Woronowicz’s theory are made. 0.1. Short description of the paper. 0.1.1. We start with constructing differential Hopf algebras (Section 1). Data for such construction are morphisms in the category of graded differential complexes. 0.1.2. Given a Hecke R-matrix for a vector space V , we construct in this paper another Hecke R-matrix R for the space W = V ⊕ V equipped with the differential d = (