Finite Dimensional Algebras and Quantum Groups

Front Cover
American Mathematical Society, 2008 - Mathematics - 759 pages
0 Reviews
"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 Ringel-Hall algebra realization for the positive part of a quantum enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the Beilinson-Lusztig-MacPherson 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 Kac-Moody Lie algebras. These two independent theories eventually meet in Part 4, under the umbrella of Ringel-Hall algebras. Cartan matrices can also be used to define an important class of groups--Coxeter groups--and their associated Hecke algebras. Hecke algebras associated with symmetric groups give rise to an interesting class of quasi-hereditary 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 Beilinson-Lusztig-MacPherson 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."--Publisher's website.

What people are saying - Write a review

We haven't found any reviews in the usual places.

Related books

Other editions - View all

About the author (2008)

Dr Bangming Deng is a Professor of Mathematics at Beijing Normal University, having previously been an Alexander von Humboldt Fellow at Bielefeld University. His main research interests are representation theory of algebras, Ringel Hall algebras, Lie algebras and quantum groups. He is a co-author of the book Finite Dimensional Algebras and Quantum Groups (American Mathematical Society, 2008) and has served as editor for the Journal of Frontiers of Mathematics in China.

Bibliographic information