On the assembly map for complex semisimple quantum groups

TL;DR

This paper proves that complex semisimple quantum groups satisfy a categorical version of the Baum-Connes conjecture, extending the understanding of their K-theoretic properties using homological algebra techniques.

Contribution

It introduces a categorical approach to the Baum-Connes conjecture for complex semisimple quantum groups, compatible with deformation methods, and defines an assembly map with arbitrary coefficients.

Findings

Complex semisimple quantum groups satisfy a categorical Baum-Connes conjecture.

The approach uses homological algebra in triangulated categories.

An assembly map with arbitrary coefficients is constructed for these quantum groups.

Abstract

We show that complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups, satisfy a categorical version of the Baum-Connes conjecture with trivial coefficients. This approach, based on homological algebra in triangulated categories, is compatible with the previously studied deformation picture of the assembly map, and allows us to define an assembly map with arbitrary coefficients for these quantum groups.