Quantum supergroups of GL(n|m) type: Differential forms, Koszul complexes, and Berezinians

1997/10/15 by Volodymyr Lyubashenko, Anthony Sudbery
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Nonlinear Waves and Solitons

paper · doi:10.1215/s0012-7094-97-09001-3

Abstract

. 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. In this paper we will be concerned with differential Hopf algebras generated by sets of matrix elements. We start (Section 1) by giving a construction of such algebras, generalising the construction [31, 27] of a bialgebra generated by a single se...

Citations

• Differential calculus on compact matrix pseudogroups (quantum groups)
• Hopf Modules and Yetter - Drinfel′d Modules

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• Introduction to Braided Geometry and q-Minkowski Space

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