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Finite Dimensional Algebras and Quantum Groups

Bangming Deng Beijing Normal University, Beijing, People’s Republic of China
Jie Du University of New South Wales, Sydney, Australia
Brian Parshall University of Virginia, Charlottesville, VA
Jianpan Wang East China Normal University, Shanghai, People’s Republic of China
Available Formats:
Hardcover ISBN: 978-0-8218-4186-0
Product Code: SURV/150
List Price: $129.00 MAA Member Price:$116.10
AMS Member Price: $103.20 Electronic ISBN: 978-1-4704-1377-4 Product Code: SURV/150.E List Price:$123.00
MAA Member Price: $110.70 AMS Member Price:$98.40
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List Price: $193.50 MAA Member Price:$174.15
AMS Member Price: $154.80 Click above image for expanded view Finite Dimensional Algebras and Quantum Groups Bangming Deng Beijing Normal University, Beijing, People’s Republic of China Jie Du University of New South Wales, Sydney, Australia Brian Parshall University of Virginia, Charlottesville, VA Jianpan Wang East China Normal University, Shanghai, People’s Republic of China Available Formats:  Hardcover ISBN: 978-0-8218-4186-0 Product Code: SURV/150  List Price:$129.00 MAA Member Price: $116.10 AMS Member Price:$103.20
 Electronic ISBN: 978-1-4704-1377-4 Product Code: SURV/150.E
 List Price: $123.00 MAA Member Price:$110.70 AMS Member Price: $98.40 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$193.50 MAA Member Price: $174.15 AMS Member Price:$154.80
• Book Details

Mathematical Surveys and Monographs
Volume: 1502008; 759 pp
MSC: 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 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.

Graduate students and research mathematicians interested in quantum groups and finite-dimensional 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. Kazhdan-Lusztig combinatorics for Hecke algebras
• 8. Cells and representations of symmetric groups
• 9. The integral theory of quantum Schur algebras
• 10. Ringel-Hall 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

• 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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Volume: 1502008; 759 pp
MSC: 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 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.

Graduate students and research mathematicians interested in quantum groups and finite-dimensional 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. Kazhdan-Lusztig combinatorics for Hecke algebras
• 8. Cells and representations of symmetric groups
• 9. The integral theory of quantum Schur algebras
• 10. Ringel-Hall 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
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