Representation Theory and Higher Algebraic K-Theory

This book provides a unified treatment of a number of deep results connecting representation theory of groups in the broad sense of (continuous) homomorphisms from a group to automorphisms of objects of a category on one hand, and higher K-theory (the structure of Kn, Gn-groups) on the other hand. The book consists of three parts. The first one reviews classical algebraic K-theory (Grothendieck groups and rings, K0, K1, K2, Kn, and their applications to orders and group rings). The second part presents the main results of higher K-theory and their applications to integral representations. The third part employs Mackey, Green, and Burnside functors in the development of equivariant higher algebraic K-theory and equivariant homology theory. The book culminates in a detailed discussion of the Baum-Connes conjecture and related conjectures on certain assembly maps being isomorphisms. Given the amount of results presented in the book, the author is forced in some cases to sketch proofs or refer to other sources for details. However, there are a number of detailed computations, especially of the Gn-groups in section 7. There are also two appendices. The first collects together important computational tools of K-theory, the second presents eighteen open problems. Written by a leading expert in the field, this monograph will be indispensable for anyone interested in contemporary representation theory and its K-theoretic aspects.

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