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《ECNU 2014 Hopf代數(shù)、量子群與表示論暑期學(xué)校及工作營》

2014.7.27---8.9, 華東師范大學(xué)數(shù)學(xué)系擬將舉辦研究主題為《Hopf代數(shù)、量子群與表示論》的為期兩周的研究生暑期學(xué)校，含2天的學(xué)術(shù)會(huì)議。Kaplansky 1975年在Hopf代數(shù)領(lǐng)域提出的10個(gè)猜想一直引領(lǐng)著該領(lǐng)域的發(fā)展，近10多年來，隨著量子群發(fā)展步入新階段，也推動(dòng)著Hopf代數(shù)領(lǐng)域尤其是分類工作的迅猛發(fā)展，其研究方法、思想手段、觀點(diǎn)看法的更新和深入日新月異，近年來最重大的突破是Andruskiewitsch-Schneider 完成了“以有限Abel群代數(shù)為余根基的有限維點(diǎn)Hopf代數(shù)分類工作”---2010年發(fā)表在國際頂級數(shù)學(xué)刊物美國數(shù)學(xué)年刊Annals Math.上，引發(fā)了新的國際研究熱點(diǎn)。

華東師大數(shù)學(xué)系胡乃紅教授為本次暑期學(xué)校邀請了工作在Hopf代數(shù)、量子群和表示論國際前沿的著名專家和學(xué)者：Hopf代數(shù)分類學(xué)權(quán)威專家來自阿根廷的Andruskiewitsch教授、量子群著名專家巴黎七大Rosso教授、澳大利亞悉尼大學(xué)著名數(shù)學(xué)物理及不變量專家張瑞斌教授（中科大千任計(jì)劃學(xué)者）、俄羅斯著名模李代數(shù)及Hopf代數(shù)專家Skryabin 研究員、美國李理論與同調(diào)代數(shù)專家Feldvoss教授，分別來講授每人8小時(shí)的短課程，共計(jì)40學(xué)時(shí)。暑期班將開設(shè)以下課程：

(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.

[課程內(nèi)容介紹]

(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.

[暑期學(xué)校規(guī)模]

本暑期學(xué)校預(yù)設(shè)的聽眾是本校代數(shù)方向的感興趣的碩士生、博士生、博士后和青年教師，以及本校畢業(yè)的相關(guān)方向的青年教師和博士后，并接受部分兄弟院校相關(guān)方向的博士研究生，總的聽眾規(guī)模約50人，接納外校報(bào)名博士生25人，住研究生公寓。

[聯(lián)系人}：華東師大數(shù)學(xué)系辦公室張紅艷 hyzhang@math.ecnu.edu.cn, (021) 54342609

[致謝]

本次暑期學(xué)校受到華東師大研究生院培養(yǎng)處暑期學(xué)校項(xiàng)目、 數(shù)學(xué)系111項(xiàng)目及國家自然科學(xué)基金項(xiàng)目等支持。特此致謝！

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