Introduction to Quantum Groups and Crystal Bases
Seiten
2002
American Mathematical Society (Verlag)
978-1-4704-7975-6 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-7975-6 (ISBN)
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The notion of a ""quantum group"" was introduced by V.G. Dinfel&dacute; and M. Jimbo, independently, in their study of the quantum Yang-Baxter equation arising from 2-dimensional solvable lattice models. Quantum groups are certain families of Hopf algebras that are deformations of universal enveloping algebras of Kac-Moody algebras. And over the past 20 years, they have turned out to be the fundamental algebraic structure behind many branches of mathematics and mathematical physics, such as solvable lattice models in statistical mechanics, topological invariant theory of links and knots, representation theory of Kac-Moody algebras, representation theory of algebraic structures, topological quantum field theory, geometric representation theory, and $C^*$-algebras. In particular, the theory of ""crystal bases"" or ""canonical bases"" developed independently by M. Kashiwara and G. Lusztig provides a powerful combinatorial and geometric tool to study the representations of quantum groups. The purpose of this book is to provide an elementary introduction to the theory of quantum groups and crystal bases, focusing on the combinatorial aspects of the theory.
Jin Hong, Korea Institute for Advanced Study, Seoul, Korea. Seok-Jin Kang, Korea Institute for Advanced Study, Seoul, Korea.
Chapters
Chapter 1. Lie algebras and Hopf algebras
Chapter 2. Kac-Moody algebras
Chapter 3. Quantum groups
Chapter 4. Crystal bases
Chapter 5. Existence and uniqueness of crystal bases
Chapter 6. Global bases
Chapter 7. Young tableaux and crystals
Chapter 8. Crystal graphs for classical Lie algebras
Chapter 9. Solvable lattice models
Chapter 10. Perfect crystals
Chapter 11. Combinatorics of Young walls
| Erscheinungsdatum | 12.06.2025 |
|---|---|
| Reihe/Serie | Graduate Studies in Mathematics |
| Verlagsort | Providence |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 1-4704-7975-3 / 1470479753 |
| ISBN-13 | 978-1-4704-7975-6 / 9781470479756 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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