An analogue of the Baum-Connes conjecture for quantum SL(2, C)

The Baum-Connes conjecture gives a description of the K-theory of the reduced group C*-algebra of a locally compact second countable group. In the case of a connected Lie group G, Connes reformulated the conjecture in terms of a deformation of G provided by a certain continuous field of C*-algebras. The conjecture is known to be true for complex semisimple Lie groups, and in 2008 Higson provided a new proof of this result, using Connes reformulation and an observation due to Mackey about the representation theories of a complex semisimple Lie group and an associated group called the Cartan motion group. In this thesis, we present and prove an analogue of the conjecture for the quantum group quantum SL(2, C) in the spirit of Connes reformulation and Higson's proof. In particular, we define a quantum version of Connes' field, which provides a deformation from quantum SL(2, C) to a quantum analogue of the Cartan motion group. We show that Mackey's observation carries over to the quantum setting, and we then prove an analogue of the conjecture using Higson's method. We also show there is compatibility between the Baum-Connes conjecture for SL(2, C) and our quantum result, in that we can construct a continuous field which encodes Connes' field and our quantum field, as well as a deformation of SL(2, C) to quantum SL(2, C) and a deformation of the Cartan motion group to the quantum Cartan motion group.