hep-th/9311095 QUANTUM SUPERGROUPS OF GL(njm) 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 de ne 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 di erential. The notion of a quantum di erential supergroup is derived. Its algebra of functions is a di erential coquasitriangular Hopf algebra, having the usual algebra of di erential 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 reconstruction theorems for di erential Hopf algebras (Section 1). Data for such reconstruction are morphisms in the category of graded di erential 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 di erential d = 0 1 0 0 and the grading : W ! W , = 1 0 0 1 . The matrix R is distinguished by the property R(d 1 + d) = (d 1 + d)R: 0.1.3. The algebra H of functions on the quantum supergroup constructed fromR is aZ-graded di erential coquasitriangular Hopf algebra (Section 2). Shortly, it de nes a di erential quantum supergroup. A quotient of H is the Z>0-graded di erential Hopf algebra of di erential forms de ned via R in [24, 25, 29, 34]. The classical version (q = 1) of this construction is: take a vector space V , add to it another copy of it with the opposite parity and consider the general linear supergroup of the obtained space.