Cartan calculus for quantum differentials on bicrossproducts

Abstract

We provide the Cartan calculus for bicovariant differential forms on bicrossproduct quantum groups \(k(M){{\blacktriangleright}\kern -2pt\triangleleft}kG\) associated to finite group factorizations X=GM and a field k. The irreducible calculi are associated to certain conjugacy classes in X and representations of isotropy groups. We find the full exterior algebras and show that they are inner by a bi-invariant 1-form θ which is a generator in the noncommutative de Rham cohomology H1. The special cases where one subgroup is normal are analysed. As an application, we study the noncommutative cohomology on the quantum codouble \(D^{*}(S_{3}){\cong}k(S_{3}){{\blacktriangleright}\kern-2pt\triangleleft }k\mathbb{Z}_{6}\) and the quantum double \(D(S_{3})=k(S_{3}){>\kern-4pt\triangleleft}kS_{3}\), finding respectively a natural calculus and a unique calculus with H0=k.1.