量子群
Starting with an associative algebra, tensor products of modules do not necessarily have a natural module structure. For bialgebras (in particular, Hopf algebras), they do. Moreover, if the bialgebra is quasitriangular then the tensor product of modules and the tensor product of the same modules taken in the opposite order are isomorphic as modules. Quantum groups, studied since the 1980s, form a rich family of quasitriangular bialgebras which deform associative algebras naturally connected with certain (e.g. semisimple finite-dimensional) Lie algebras.
讲师
日期
2022年09月20日 至 12月15日
网站
修课要求
Basic Lie algebra background. An interest or experience in mathematical physics is helpful, as well as familiarity with representation theory of groups and associative algebras.
课程大纲
Tentatively, we will discuss the following topics:
1) Preliminaries: definition and basic properties of bi- and Hopf algebras; monoidal categories; cocommutative examples (group algebras, universal enveloping algebras)
2) Quasitriangularity; braided monoidal categories; diagrammatical calculus; basic example: Sweedler's Hopf algebra
3) Chevalley-Serre presentation of finite-dimensional semisimple Lie algebras, symmetrizable Kac-Moody algebras, BGG/Kac category O
4) Main example: Drinfeld-Jimbo quantum groups U_q(g) defined over C(q) (starting with quantum sl2)
5) Quantized universal enveloping algebras U_q(g) as topological Hopf algebras; precise relation to U(g).
6) Quasitriangular structure of Drinfeld-Jimbo quantum groups, completion via Tannakian approach (natural transformations of forgetful functor)
7) Pairings between Hopf algebras; bar involution, skew derivations, quasi-R-matrix (Lusztig approach to R); quantum double (Drinfeld approach to R)
8) Explicit formulas of R for quantum sl2; factorization of R-matrices via quantum Weyl group
9) Dual quantum group: quantization of the algebra of functions (RTT relation)
10) Affine quantum groups and applications to quantum integrability
1) Preliminaries: definition and basic properties of bi- and Hopf algebras; monoidal categories; cocommutative examples (group algebras, universal enveloping algebras)
2) Quasitriangularity; braided monoidal categories; diagrammatical calculus; basic example: Sweedler's Hopf algebra
3) Chevalley-Serre presentation of finite-dimensional semisimple Lie algebras, symmetrizable Kac-Moody algebras, BGG/Kac category O
4) Main example: Drinfeld-Jimbo quantum groups U_q(g) defined over C(q) (starting with quantum sl2)
5) Quantized universal enveloping algebras U_q(g) as topological Hopf algebras; precise relation to U(g).
6) Quasitriangular structure of Drinfeld-Jimbo quantum groups, completion via Tannakian approach (natural transformations of forgetful functor)
7) Pairings between Hopf algebras; bar involution, skew derivations, quasi-R-matrix (Lusztig approach to R); quantum double (Drinfeld approach to R)
8) Explicit formulas of R for quantum sl2; factorization of R-matrices via quantum Weyl group
9) Dual quantum group: quantization of the algebra of functions (RTT relation)
10) Affine quantum groups and applications to quantum integrability
参考资料
Title A Guide to Quantum Groups
Authors V. Chari and A. Pressley
Publisher Cambridge University Press, 1995
ISBN 0521558840, 9780521558846
Title Lectures on Quantum Groups
Author J. Jantzen
Publisher American Mathematical Society
ISBN 0821872346, 9780821872345
Authors V. Chari and A. Pressley
Publisher Cambridge University Press, 1995
ISBN 0521558840, 9780521558846
Title Lectures on Quantum Groups
Author J. Jantzen
Publisher American Mathematical Society
ISBN 0821872346, 9780821872345
听众
Graduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Bart Vlaar于2022年9月以副研究员身份全职入职BIMSA。他的研究兴趣包括代数和表示论，以及它们在数学物理上的应用。他在苏格兰格拉斯哥大学获得博士学位，之后先后在阿姆斯特丹大学、诺丁汉大学、约克大学和苏格兰赫瑞瓦特大学任职位，并访问位于波恩的马斯克博朗克数学研究所。