Free shipping on orders over $99
Finite Dimensional Algebras and Quantum Groups

Finite Dimensional Algebras and Quantum Groups

by Jianpan WangBangming Deng Brian Parshall and others
Hardback
Publication Date: 30/12/2008

Share This Book:

 
$183.70
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.
ISBN:
9780821841860
9780821841860
Category:
Algebra
Format:
Hardback
Publication Date:
30-12-2008
Publisher:
American Mathematical Society
Country of origin:
United States
Pages:
763
Weight:
1.56kg

Click 'Notify Me' to get an email alert when this item becomes available

Reviews

Be the first to review Finite Dimensional Algebras and Quantum Groups.