| 标题 | 华东师范大学”Hopf代数量子群与表示论“2014年暑期学校招生简章 |
| 内容 | 《ECNU 2014 Hopf代数、量子群与表示论暑期学校及工作营》 2014.7.27---8.9, 华东师范大学数学系拟将举办研究主题为《Hopf代数、量子群与表示论》的为期两周的研究生暑期学校，含2天的学术会议。Kaplansky 1975年在Hopf代数领域提出的10个猜想一直引领着该领域的发展，近10多年来，随着量子群发展步入新阶段，也推动着Hopf代数领域尤其是分类工作的迅猛发展，其研究方法、思想手段、观点看法的更新和深入日新月异，近年来最重大的突破是Andruskiewitsch-Schneider 完成了“以有限Abel群代数为余根基的有限维点Hopf代数分类工作”---2010年发表在国际顶级数学刊物美国数学年刊Annals Math.上，引发了新的国际研究热点。 华东师大数学系胡乃红教授为本次暑期学校邀请了工作在Hopf代数、量子群和表示论国际前沿的著名专家和学者：Hopf代数分类学权威专家来自阿根廷的Andruskiewitsch教授、量子群著名专家巴黎七大Rosso教授、澳大利亚悉尼大学著名数学物理及不变量专家张瑞斌教授（中科大千任计划学者）、俄罗斯著名模李代数及Hopf代数专家Skryabin 研究员、美国李理论与同调代数专家Feldvoss教授，分别来讲授每人8小时的短课程，共计40学时。暑期班将开设以下课程： (1) On Classification of Pointed Hopf algebras, by N. Andruskiewitsch; (2) Cofree Hopf algebras and quantum groups, by M. Rosso; (3) Hopf algebras and their Actions, by S. Skryabin; (4) Introduction to Lie superalgebras and their representations, by R.B. Zhang (5) Introduction to Support Varieties and Applications, by J. Feldvoss. [课程内容介绍] (1) On Classification of Pointed Hopf algebras 1. Nichols algebras. The braid equation. Braided vector spaces and Yetter-Drinfeld modules. Alternative definitions of Nichols algebras. Basic examples. Approximations of Nichols algebras. 2. Nichols algebra of diagonal type. The PBW theorem of Kharchenko. The classification of Heckenberger. Relations with Lie (super) algebras. 3. The Weyl groupoid. Coxeter groupoids. Crystallographic data and Weyl groupoid data. Outline of the classification. Convex orders. 4. Defining relations for Nichols algebra of diagonal type. Convex orders. Quantum Serre relations and their generalizations; powers of root vectors relations. 5. The lifting method. Hopf algebras generated by the coradical. The coradical filtration and the standard filtration. The associated graded Hopf algebras. Bosonization and the role of Nichols algebras. 6. Deformations of Nichols algebras. The general strategy. Classification results for pointed Hopf algebras with abelian group. 7. Nichols algebra of group type. Racks and cocycles. Classification of simple racks. Examples of finite-dimensional Nichols algebras of group type. The Fomin-Kirillov algebras. 8. Pointed Hopf algebras with non-abelian group. The collapsing criteria. Classification results for pointed Hopf algebras with non-abelian group. (2) Cofree Hopf algebras and quantum groups Abstract: Connected cofree Hopf algebras have a universal property allowing the construction of many compatible Hopf algebra structures. They were classified by J-L Loday and M. Ronco, and familiar examples include shuffle Hopf algebras and quasi-shuffle Hopf algebras which appear in many domains of mathematics: combinatorics, number theory (multiple zeta values), Rota-Baxter algebras, ... Replacing the ground field by a Hopf algebra H leads to a wide extension of the framework; the relevant category is that of Hopf bimodules M over H, and for each M, one can associate a natural (not connected) cofree coalgebra, first introduced by W. Nichols. The classification of compatible Hopf algebra structures leads, in particular examples, to quantum quasi shuffle algebras and to a new construction of quantized envelopping algebras. This provides a new framework to construct representations. (3) Hopf algebras and their Actions Abstract. Hopf algebras have found important applications in various areas of mathematics. At the same time the structural properties of Hopf Algebras remain far from being fully understood.This series of lectures will start at the basics of the theory and will move gradually on towards deeper results describing ring-theoretic properties of Hopf algebras, their actions and coactions on associative algebras. Particular questions discussed are conditions ensuring that a Hopf module algebra is Frobenius or quasi-Frobenius, existence of classical quotient rings for Hopf module algebras, extension of the module structure to quotient rings, projectivity and faithful flatness of Hopf algebras over Hopf subalgebras and right coideal subalgebras. An important tool in the study are equivariant and coequivariant modules. (4) Introduction to Lie superalgebras and their representations Outline： 1. Lie superalgebras The general linear Lie superalgebra gl(m|n), orthosymplectic Lie superalgebra osp(m|2n); simple Lie superalgebras of classical type. 2. Invariant theory Tensor representations of gl(m|n), first fundamental theorem of invariant theory for gl(m|n), Schur-Weyl duality, a super duality; tensor representations of osp(m|2n), Schur-Weyl-Brauer duality. 3. Parabolic category O of gl(m|n) Parabolic category O; canonical bases of quantum gl(∞); Kazhan-Lusztig polynomials of gl(m|n); a closed character formula and dimension formula; Jantzen filtration for Kac modules. 4. Finite dimensional representations of osp(m|2n) Flag supermanifolds; elements of Bott-Borel-Weil theory for osp(m|2n); a combinatorial algorithm for computing characters. (5) Introduction to Support Varieties and Applications Abstract: In this series of lectures I will start out by defining the complexity of a finite dimensional module over a self-injective algebra and prove its main properties. All this is well-known for group algebras of finite groups, or more generally of finite group schemes, but is valid in this more general context. Then I will explain that for certain algebras over an algebraically closed ground field the complexity of a module can be realized as the dimension of an affine variety, the so-called support variety of the module. I will describe several properties of support varieties and I will present several applications of these concepts which were introduced originally for modular representations of _nite groups by Alperin and Carlson in the late seventies and the early eighties. In particular, I will define the representation type of an associative algebra and state the trichotomy theorem of Drozd. Then I will explain how support varieties can be used to prove a \theorem" of Rickard for self-injective algebras with finite cohomology. Finally, this will be applied to reduced enveloping algebras of restricted Lie algebras, to small quantum groups (a proof of Cibils' conjecture), and if time permits to Hecke algebras of classical type. [暑期学校规模] 本暑期学校预设的听众是本校代数方向的感兴趣的硕士生、博士生、博士后和青年教师，以及本校毕业的相关方向的青年教师和博士后，并接受部分兄弟院校相关方向的博士研究生，总的听众规模约50人，接纳外校报名博士生25人，住研究生公寓。 [联系人}：华东师大数学系办公室张红艳 hyzhang@math.ecnu.edu.cn, (021) 54342609 [致谢] 本次暑期学校受到华东师大研究生院培养处暑期学校项目、 数学系111项目及国家自然科学基金项目等支持。特此致谢！ |
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