Complex quantum groups and a deformation of the Baum-Connes assembly map

TL;DR

This paper extends the Baum-Connes assembly map to complex semisimple quantum groups, establishing an isomorphism that links quantum group K-theory with classical topology through a deformation framework.

Contribution

It introduces a quantum analogue of the Baum-Connes assembly map for complex semisimple quantum groups and proves its isomorphism, connecting quantum and classical K-theory.

Findings

Quantum assembly map is an isomorphism.

Classical Baum-Connes map is a direct summand of the quantum map.

A continuous field of C*-algebras encodes both quantum and classical maps.

Abstract

We define and study an analogue of the Baum-Connes assembly map for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups. Our starting point is the deformation picture of the Baum-Connes assembly map for a complex semisimple Lie group $ G $, which allows one to express the $ K $-theory of the reduced group $ C^* $-algebra of $ G $ in terms of the $ K $-theory of its associated Cartan motion group. The latter can be identified with the semidirect product of the maximal compact subgroup $ K $ acting on $ \mathfrak{k}^* $ via the coadjoint action. In the quantum case the role of the Cartan motion group is played by the Drinfeld double of the classical group $ K $, whose associated group $ C^* $-algebra is the crossed product of $ C(K) $ with respect to the adjoint action of $ K $. Our quantum assembly map is obtained by varying the deformation parameter in the Drinfeld double construction applied to the standard deformation $ K_q $ of $ K $. We prove that the quantum assembly map is an isomorphism, thus providing a description of the $ K $-theory of complex quantum groups in terms of classical topology. Moreover, we show that there is a continuous field of $ C^* $-algebras which encodes both the quantum and classical assembly maps as well as a natural deformation between them. It follows in particular that the quantum assembly map contains the classical Baum-Connes assembly map as a direct summand.