Hardcover ISBN:  9780821841860 
Product Code:  SURV/150 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
Electronic ISBN:  9781470413774 
Product Code:  SURV/150.E 
List Price:  $123.00 
MAA Member Price:  $110.70 
AMS Member Price:  $98.40 

Book DetailsMathematical Surveys and MonographsVolume: 150; 2008; 759 ppMSC: Primary 05; 16; 17; 20;
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 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 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.ReadershipGraduate students and research mathematicians interested in quantum groups and finitedimensional algebras.

Table of Contents

Chapters

0. Getting started

1. Representations of quivers

2. Algebras with Frobenius morphisms

3. Quivers with automorphisms

4. Coxeter groups and Hecke algebras

5. Hopf algebras and universal enveloping algebras

6. Quantum enveloping algebras

7. KazhdanLusztig combinatorics for Hecke algebras

8. Cells and representations of symmetric groups

9. The integral theory of quantum Schur algebras

10. RingelHall algebras

11. Bases of quantum enveloping algebras of finite type

12. Green’s theorem

13. Serre relations in quantum Schur algebras

14. Constructing quantum $\mathrm {gl}_n$ via quantum Schur algebras


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...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.
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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 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 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.
Graduate students and research mathematicians interested in quantum groups and finitedimensional algebras.

Chapters

0. Getting started

1. Representations of quivers

2. Algebras with Frobenius morphisms

3. Quivers with automorphisms

4. Coxeter groups and Hecke algebras

5. Hopf algebras and universal enveloping algebras

6. Quantum enveloping algebras

7. KazhdanLusztig combinatorics for Hecke algebras

8. Cells and representations of symmetric groups

9. The integral theory of quantum Schur algebras

10. RingelHall algebras

11. Bases of quantum enveloping algebras of finite type

12. Green’s theorem

13. Serre relations in quantum Schur algebras

14. Constructing quantum $\mathrm {gl}_n$ via quantum Schur algebras

...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