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

2018; American Mathematical Society; Volume: 371; Issue: 12 Linguagem: Inglês

10.1090/tran/7774

1088-6850

Andrew Monk, Christian Voigt,

Noncommutative and Quantum Gravity Theories

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.