Classification of differentials and Cartan calculus on bicrossproducts

Details Classification of differentials and Cartan calculus on bicrossproducts Classification of differentials and Cartan calculus on bicrossproducts

2002-05-17

arxiv.org/abs/math/0205194v2

Mathematics

Quantum Algebra

Minor revision: deleted six pages (mainly some examples) at request of referee. Now 30 pages latex, no figs

Scientific paper

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 $\theta$ which is a generator in the noncommutative de Rham cohomology $H^1$. The special cases where one subgroup is normal are analysed. As an application, we study the noncommutative cohomology on the quantum codouble $D^*(S_3)\isom k(S_3)\lrbicross k\Z_6$ and the quantum double $D(S_3)=k(S_3)\lcross kS_3$, finding respectively a natural calculus and a unique calculus with $H^0=k.1$.

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