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An Isaac Newton Institute Workshop

Noncommutative Geometry and Cyclic Cohomology

On the Hochschild homology of quantum groups

Author: Ulrich Kraehmer (IMPAN Warsaw)


This talk reports on two papers written in collaboration with T.Hadfield. In the first we computed the Hochschild homology of the standard quantisation of SL(2) with coefficients in bimodules obtained from the algebra itself by twisting the multiplication on one side by an automorphism. It turned out that precisely for Woronowicz's modular automorphism there is a unique nontrivial Hochschild class in degree 3=dim(SL(2)), in contrast to the untwisted case where all homologies were known to vanish in degrees greater than one. In the second paper we generalised this to SL(N) by showing that the quantisations satisfy van den Bergh's analogue of Poincare duality in Hochschild (co)homology. This was extended recently by K.Brown and J.Zhang to all Noetherian Artin-Schelter Gorenstein Hopf algebras. These results clarify the purely homological relevance of the twisted coefficients whose study was originally motivated by their relation to the theory of covariant differential calculi over quantum groups.