Summer School on Quantum Groups and Lie Theory in ECNU 20082008华东师大数学系研究生暑期学校《量子群和李理论海外专家系列讲学活动》（2008年7月10日---7月24日）
学生报到地点：理科大楼A1502 ，报到时间：9:00-16:00
Organizer: Prof. Dr. Naihong HU Lecture (I) ---Nichols Algebras of Diagonal Type by I. Heckenberger (德国慕尼黑) Abstract: The theory of Nichols algebras of diagonal type has its origins in a work of W. Nichols from 1978 as he tried to classify a class of finite dimensional Hopf algebras. Since the discovery of quantum groups by Drinfel’d and Jimbo it is known that Nichols algebras are closely related to the theory of semisimple Lie algebras and Lie superalgebras. Later it turned out that all Nichols algebras of diagonal type hold a combinatorics which is a nontrivial generalization of the combinatorics of Weyl groups and associated root systems, and these structures admit a full classification of (say finite dimensional) Nichols algebras of diagonal type. The knowledge of the structure of Nichols algebras of diagonal type is crucial for the method of Andruskiewitsch and Schneider to classify all pointed Hopf algebras with finite Gelfand-Kirillov dimension and abelian coradical. In this course an introduction to Nichols algebras (of diagonal type) and their combinatorics is given, and some examples are presented. The classification scheme is sketched, and further related constructions like Drinfel’d doubles, Lusztig isomorphisms are explained. 课程目录：（每讲2小时） 1. Hopf algebras and braided Hopf algebras2. Nichols algebras3. Kharchenko's PBW basis of Nichols algebras of diagonal type4. Weyl groupoids5. Root systems of Nichols algebras of diagonal type6. Classification of finite Nichols algebras of diagonal type7. Drinfeld doubles of Nichols algebras of diagonal type and Lusztig isomorphisms8. Outlook
Lecture (II)--- The Tensor Product Theory for Modules for Affine Lie Algebras of Fixed Levels By Huang Yi-Zhi (美国Rutgers) Abstract: For a suitable category of modules for an affine Lie algebra of a negative level, Kazhdan and Lusztig first constructed a rigid braided tensor category structure on the category. In the context of the representation theory of vertex operator algebras, Lepowsky and I constructed, among many other things, a braided tensor category structure on the category of finite direct sums of integrable highest weight modules for an affine Lie algebra of a positive integral level. Using the verlinde conjecture I proved, I proved the rigidity and the non-degeneracy property of the braided tensor category. Recently Lepowsky, Zhang and I developed a logarithmic tensor product theory which includes the theory of Kazhdan-Lusztig and the early theory of Lepowsky and I mentioned above as special cases. In these lectures, I will explain this general tensor product theory in the special case of affine Lie algebras.
课程目录：（每讲2小时） 1. Affine Lie algebras and modules 2. Vertex operator algebras associated to affine Lie algebras,modules and (logarithmic) intertwining operators 3. Definitions of braided tensor category and modular tensor category 4. Tensor product bifunctor and its constructions 5. Differential equations of regular singularities 6. Associativity and commutativity isomorphisms 7. The coherence properties, rigidity and nondegeneracy property Lecture (III)--- Introduction to Extended Affine Lie Algebras By Gao Yun (加拿大约克) Abstract: Extended affine Lie algebras are a natural generalization of affine Kac-Moody Lie algebras. They are closely related to the extended affine root systems of K.Saito, interesction matrix Lie algebras of P.Slodowy, and root graded Lie algebras studied by Berman-Moody, Benkart-Zelmanov, Neher, Allison-Benkart-Gao, Benkart-Smirnov. This newly developed Lie algebras include toroidal Lie algebras as examples. In this series of lectures, I will give definitions and many examples. Then I will show how those Lie algebras can be classified by relating with the extended affine root systems and using the root graded Lie algebras. Finally I will provide some module constructions for some of extended affine Lie algebras. 课程目录：（每讲2小时） 1. Definitions of EALAs 2. Examples of EALAs 3. Classifications: Root systems 4. Classifications: Lie algebras I--Associative coordinates 5. Classifications: Lie algebras II--Alternative coordinates 6. Classifications: Lie algebras III--Jordan coordinates 7. Representation theory: Vertex operator construction 8. Representation theory: Boson and fermi construction
Lecture (IV)--- Vertex Representations of Affine Lie Algebras and Generalizations By Jing Naihuan (美国北卡) Abstract: Affine Lie algebras are the most important examples of Kac-Moody Lie algebras, and many of the finite-dimensional Lie theory can be generalized to the infinite dimensional case. In this course we will discuss some of the special features of affine Lie algebras and their representations. We will start with the simplest example of the affine Lie algebra sl^(2) to explain its two commonly used vertex operator realizations: the homogeneous and principal picture. We then move forward to more general affine Lie algebras and present its vertex representations (mostly level one). In the end we hope to briefly discuss some of the generalizations such as quantum groups and vertex (operator) algebras. This introductory short course consists of eight lectures, and a list of the detailed topics are as follows. 课程目录：（每讲2小时） 1) Affine Lie algebras Kac-Moody Lie algebras, realizations of affine Lie algebras 2) Weyl groups, affine Weyl groups 3) Representations of Affine Lie algebras Category O, and general theory of highest weight modules, Verma modules, Character formula 4) Heisenberg algebras Uniqueness of irreducible representations, identification with differential operators on the polynomial ring. 5) Homogeneous vertex representations of affine Lie algebras Representations of the affine Lie algebra sl^(2), generalization to ADE cases 6) Principal vertex representations of affine Lie algebras Examples of classical affine Lie algebras sl^(n) 7) Sugawara operators and representations of the Virasoro algebra 8) Generalizations to quantum affine Lie algebras, vertex operator algebras
Lecture (V)---Hopf Algebras with Trace and Representation By Marc Rosso (法国巴黎高师) Abstract: We study the restriction of representations of Cayley-Hamilton algebras to subalgebras. This theory is applied to determine tesor products and branching rules for representation of quantum groups at roots of 1.
课程目录：（每讲1小时） 1）n-dimensional representations 2）Cayley-Hamilton algebras 3）Semisimple representations 4）The reduced trace 5）The unramified locus and restriction maps 6）Quantized universal enveloping algebras at roots of 1 7）Clebsch-Gordan decompositions for generic representations of quantized universal enveloping algebras at roots of 1.
Lecture (VI)---Representations and Cohomology for Lie Superalgebras By Jonathan Kujawa (Oklahoma) Abstract: In this series of lectures we will provide an introduction to some recent developments in the representation theory of Lie superalgebras. We will articularly emphasize recent work of ours (in collaboration with Bagci, Boe, and Nakano) on cohomology and support varieties. In particular, we will see that we are able to adapt tools from finite groups in positive characteristic to obtain new insights into the characteristic 0 theory of Lie superalgebras. We also will discuss recent work of ours and others which shows that the combinatorics of crystals (in the sense of Kashiwara) controls the representation theory of Lie superalgebras. The topics and depth of the talks will depend on the audience's interests and background. Every effort will be made to make the lectures accessible all attendees.
课程目录：（每讲1小时） 1) Preliminary to Lie superalgebras 2) Cohomology and support varieties3) Crystals in super cases4) Recent advances in representations of Lie superalgebras (I)5) Recent advances in representations of Lie superalgebras (II)
7月23日至24日为两天的短会---《量子群与李理论会议》 （拟安排12个报告）
致谢： 本次暑期班活动受到学校国际交流处、研究生院、数学系111引智计划项目、数学系教育部“代数几何与表示论”创新团队项目以及景乃桓与胡乃红的合作项目杰青（B类）的支持。
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