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

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.Comment: 26 page