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【内容】
“量子群”的概念是 V.G. Drinfel'd 和 M. Jimbo 在各自研究由二维可解格模型得到的量子Yang-Baxter方程时独立引入的。量子群是Hopf代数的某些族，这些族是Kac-Moody代数的泛包络代数的变形。在过去的三十年中，它们已成为数学和数学物理的许多分支背后的基本代数结构，例如统计力学中的可解格模型，链环和结点的拓扑不变理论，Kac-Moody代数的表示论，代数结构的表示论，拓扑量子场论，几何表示论和C*-代数。
特别地，由 M. Kashiwara 和 G. Lusztig 独立发展的“晶体基”或“典范基”理论为研究量子群的表示提供了一种强大的组合和几何工具。本书的目的是提供量子群和晶体基理论的基本介绍，重点放在理论的组合方面。
本书适合对非结合环和代数感兴趣的研究生阅读，也可供相关研究人员参考。
特别地，由 M. Kashiwara 和 G. Lusztig 独立发展的“晶体基”或“典范基”理论为研究量子群的表示提供了一种强大的组合和几何工具。本书的目的是提供量子群和晶体基理论的基本介绍，重点放在理论的组合方面。
本书适合对非结合环和代数感兴趣的研究生阅读，也可供相关研究人员参考。
【目录】
Introduction
Chapter 1. Lie Algebras and Hopf Algebras
1.1. Lie algebras
1.2. Representations of Lie algebras
1.3. The Lie algebra 8[(2, F)
1.4. The special linear Lie algebra [(n, F)
1.5. Hopf algebras
Exercises
Chapter 2. Kac-Moody Algebras
2.1. Kac-Moody algebras
2.2. Classification of generalized Cartan matrices
2.3. Representation theory of Kac-Moody algebras
2.4. The category Oint
Exercises
Chapter 3. Quantum Groups
3.1. Quantum groups
3.2. Representation theory of quantum groups
3.3. Al-forms
3.4. Classical limit
3.5. Complete reducibility of the category oiqt
Exercises
Chapter 4. Crystal Bases
4.1. Kashiwara operators
4.2. Crystal bases and crystal graphs
4.3. Crystal bases for Uq(12)-modules
4.4. Tensor product rule
4.5. Crystals
Exercises
Chapter 5. Existence and Uniqueness of Crystal Bases
5.1. Existence of crystal bases
5.2. Uniqueness of crystal bases
5.3. Kashiwara's grand-loop argument
Exercises
Chapter 6. Global Bases
6.1. Balanced triple
6.2. Global basis for V(A)
6.3. Polarization on Uq (g)
6.4. Triviality of vector bundles over p1
6.5. Existence of global bases
Exercises
Chapter 7. Young Tableaux and Crystals
7.1. The quantum group Uq(ln)
7.2. The category O>
7.3. Tableaux and crystals
7.4. Crystal graphs for Uq(gIn)-modules
Exercises
Chapter 8. Crystal Graphs for Classical Lie Algebras
8.1. Example: Uq(B3)-crystals
8.2. Realization of Uq(An-1)-crystals
8.3. Realization of Uq(Cn)-crystals
8.4. Realization of Uq(Bn)-crystals
8.5. Realization of Uq(Dn)-crystals
8.6. Tensor product decomposition of crystals
Exercises
Chapter 9. Solvable Lattice Models
9.1. The 6-vertex model
9.2. The quantum affine algebra Uq('[2)
9.3. Crystals and paths
Exercises
Chapter 10. Perfect Crystals
10.1. Quantum affine algebras
10.2. Energy functions and combinatorial R-matrices
10.3. Vertex operators for Uq([2)-modules
10.4. Vertex operators for quantum affine algebras
10.5. Perfect crystals
10.6. Path realization of crystal graphs
Exercises
Chapter 11. Combinatorics of Young Walls
11.1. Perfect crystals of level 1 and path realization
11.2. Combinatorics of Young walls
11.3. The crystal structure
11.4. Crystal graphs for basic representations
Exercises
Bibliography
Index of symbols
Index
Chapter 1. Lie Algebras and Hopf Algebras
1.1. Lie algebras
1.2. Representations of Lie algebras
1.3. The Lie algebra 8[(2, F)
1.4. The special linear Lie algebra [(n, F)
1.5. Hopf algebras
Exercises
Chapter 2. Kac-Moody Algebras
2.1. Kac-Moody algebras
2.2. Classification of generalized Cartan matrices
2.3. Representation theory of Kac-Moody algebras
2.4. The category Oint
Exercises
Chapter 3. Quantum Groups
3.1. Quantum groups
3.2. Representation theory of quantum groups
3.3. Al-forms
3.4. Classical limit
3.5. Complete reducibility of the category oiqt
Exercises
Chapter 4. Crystal Bases
4.1. Kashiwara operators
4.2. Crystal bases and crystal graphs
4.3. Crystal bases for Uq(12)-modules
4.4. Tensor product rule
4.5. Crystals
Exercises
Chapter 5. Existence and Uniqueness of Crystal Bases
5.1. Existence of crystal bases
5.2. Uniqueness of crystal bases
5.3. Kashiwara's grand-loop argument
Exercises
Chapter 6. Global Bases
6.1. Balanced triple
6.2. Global basis for V(A)
6.3. Polarization on Uq (g)
6.4. Triviality of vector bundles over p1
6.5. Existence of global bases
Exercises
Chapter 7. Young Tableaux and Crystals
7.1. The quantum group Uq(ln)
7.2. The category O>
7.3. Tableaux and crystals
7.4. Crystal graphs for Uq(gIn)-modules
Exercises
Chapter 8. Crystal Graphs for Classical Lie Algebras
8.1. Example: Uq(B3)-crystals
8.2. Realization of Uq(An-1)-crystals
8.3. Realization of Uq(Cn)-crystals
8.4. Realization of Uq(Bn)-crystals
8.5. Realization of Uq(Dn)-crystals
8.6. Tensor product decomposition of crystals
Exercises
Chapter 9. Solvable Lattice Models
9.1. The 6-vertex model
9.2. The quantum affine algebra Uq('[2)
9.3. Crystals and paths
Exercises
Chapter 10. Perfect Crystals
10.1. Quantum affine algebras
10.2. Energy functions and combinatorial R-matrices
10.3. Vertex operators for Uq([2)-modules
10.4. Vertex operators for quantum affine algebras
10.5. Perfect crystals
10.6. Path realization of crystal graphs
Exercises
Chapter 11. Combinatorics of Young Walls
11.1. Perfect crystals of level 1 and path realization
11.2. Combinatorics of Young walls
11.3. The crystal structure
11.4. Crystal graphs for basic representations
Exercises
Bibliography
Index of symbols
Index
