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Classification of differentials and Cartan calculus on bicrossproducts

F. Ngakeu·S. Majid·J-P. Ezin
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Abstract

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 $H^1$. 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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F. Ngakeu, S. Majid, J-P. Ezin. 2003-10-09. Classification of differentials and Cartan calculus on bicrossproducts. https://arxiv.org/abs/math/0205194

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