
Preface  Preview Material  Table of Contents  Supplementary Material 
Mathematical Surveys and Monographs 2008; 759 pp; hardcover Volume: 150 ISBN10: 0821841866 ISBN13: 9780821841860 List Price: US$123 Member Price: US$98.40 Order Code: SURV/150 See also: Lectures on Quantum Groups  Jens Carsten Jantzen Representations of Quantum Algebras and Combinatorics of Young Tableaux  Susumu Ariki Introduction to Quantum Groups and Crystal Bases  Jin Hong and SeokJin Kang Geometric Representation Theory and Extended Affine Lie Algebras  Erhard Neher, Alistair Savage and Weiqiang Wang  The interplay between finite dimensional algebras and Lie theory dates back many years. In more recent times, these interrelations have become even more strikingly apparent. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. More precisely, it investigates the RingelHall algebra realization for the positive part of a quantum enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the BeilinsonLusztigMacPherson realization for the entire quantum \(\mathfrak {gl}_n\). The book begins with the two realizations of generalized Cartan matrices, namely, the graph realization and the root datum realization. From there, it develops the representation theory of quivers with automorphisms and the theory of quantum enveloping algebras associated with KacMoody Lie algebras. These two independent theories eventually meet in Part 4, under the umbrella of RingelHall algebras. Cartan matrices can also be used to define an important class of groupsCoxeter groupsand their associated Hecke algebras. Hecke algebras associated with symmetric groups give rise to an interesting class of quasihereditary algebras, the quantum Schur algebras. The structure of these finite dimensional algebras is used in Part 5 to build the entire quantum \(\mathfrak{gl}_n\) through a completion process of a limit algebra (the BeilinsonLusztigMacPherson algebra). The book is suitable for advanced graduate students. Each chapter concludes with a series of exercises, ranging from the routine to sketches of proofs of recent results from the current literature. Readership Graduate students and research mathematicians interested in quantum groups and finitedimensional algebras. Reviews "...prove[s] to be a valuable reference to researchers working in the field. It contains and collects many results which have not appeared before in book form."  Mathematical Reviews 


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