Twisted homology of quantum groups

Tom Hadfield, Queen Mary, University of London

I will describe recent work with Ulrich Kraehmer (Warsaw), showing the dimension drop in Hochschild homology for standard quantum SL(N) is overcome via the twisted Hochschild homology of Kustermans, Murphy and Tuset.

The motivation for this work arises from the application of Alain Connes' noncommutative geometry to quantum groups, particularly in the operator-algebraic setting. Cyclic cohomology plays a central role in Connes' theory, with the simplest route to cyclic cocycles coming via so-called differential calculi. It was shown by Feng and Tsygan that many standard quantum groups suffer a "dimension drop" in Hochschild homology as we pass from the commutative to the noncommutative case, and this was for a long time thought to indicate that quantum groups were somehow incompatible with Connes' theory.

In C*-algebraic setting of compact quantum groups, so-called bicovariant differential calculi were extensively studied by Woronowicz, who showed that these gave rise to "twisted" , rather than ordinary, cyclic cocycles. Kustermans, Murphy and Tuset showed that this gave rise to a new cyclic theory, which they called twisted cyclic cohomology. In joint work with Ulrich Kraehmer, we use results of van den Bergh to show that the dimension drop in Hochschild homology is overcome for quantum SL(N) by passing to coefficients in an appropriate twisted bimodule, and we discuss extensions of this to larger classes of quantum groups.