Abstract
We provide the Cartan calculus for bicovariant differential forms on bicrossproduct quantum groups k(M)
k G 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)≅k(S 3)
kℤ6 and the quantum double D(S 3)\( > \triangleleft \) k S 3, finding respectively a natural calculus and a unique calculus with H 0 = k.1.
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Ngakeu, F., Majid, S. & Ezin, J.P. Cartan calculus for quantum differentials on bicrossproducts. Acta Appl Math 84, 193–236 (2004). https://doi.org/10.1007/s10440-005-4592-5
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DOI: https://doi.org/10.1007/s10440-005-4592-5