2002/05/17 by F. Ngakeu, Ngakeu, F., S. Majid +3
Mathematics · #58B32 #58B34 #Differential Geometry (math.DG) #FOS: Mathematics #Group Theory (math.GR) #Quantum Algebra (math.QA) #math.DG #math.GR #math.QA #msc:58B32 #msc:58B34
paper · pdf · doi:10.48550/arxiv.math/0205194
Minor revision: deleted six pages (mainly some examples) at request of referee. Now 30 pages latex, no figs
arxiv created 2003/10/09 · arxiv updated 2009/11/30
We provide the Cartan calculus for bicovariant differential forms on bicrossproduct quantum groups k(M)\lrbicross 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^*(S3)\isom k(S3)\lrbicross k\Z6 and the quantum double D(S3)=k(S3)\lcross kS3, finding respectively a natural calculus and a unique calculus with H0=k.1.