Sino-US Workshop and Summer School on

Quantum Groups,

Representation Theory and Related Topics

PROGRAM

 


 

July 11 - July 26, 2010

Department of Mathematics

East China Normal University

Shanghai, China

 

Venue of Summer School and Workshop

（I）2rd Teaching Building, Lecture Room 104 （第二教学楼104）for Summer School (July 11---22)，Minhang Campus, East China Normal University (ECNU) （华东师大闵行校区）

（II）Jiaoshi Zhi Jia Building, Lecture Hall 321 （教师之家3楼多功能厅321）for Workshop  (July 24---26)，Hongmei South Road 5858 （虹梅南路5858号）,  Minhang District, ECNU （华东师大闵行校区）

Part I. Summer School

1. Lecturers

.V. Chari (C): 8 Lectures

.Y. Gao (G): 8 Lectures

. S. Kang (K): 8 Lectures

. O. Mathieu (M): 8 Lectures

. N. Reshetikhin (R): 8 Lectures

. R. Zhang (Z): 9 Lectures

2. Lecture Schedule of Summer School

(Lecture Room 104,

2rd Teaching Building 第二教学楼)

10

Sat

11

Sun

12

Mon

13

Tue

14

Wed

15

Thu

16

Fri

8:20-9:10

Free

Free

Z

M

Z

Z

Z

9:20-10:10

Z

M

Z

M

M

10:25-11:15

RP (Y. Yao)

RP (H. Zhang)

Z

M

QA

(M)

11:15-12:15

Lunch Time

15:00-15:50

C

M

C

C

C

Z

16:10-17:00

C

M

C

C

QA

(C)

QA

(Z)

QA=Question Anwser

(Lecture Room 104, 2rd Teaching Building 第二教学楼)

17

Sat

18

Sun

19

Mon

20

Tue

21

Wed

22

Thu

23

Fri

8:20-9:10

Free

Free

R

K

G

G

World

Expo

Visit

 

9:20-10:10

R

K

R

K

10:25-11:15

R

K

QA

(R)

QA

(K)

11:15-12:00

Lunch Time

13:50-14:40

K

R

G

G

14:45-16:35

K

R

G

G

16:40-17:30

K

R

G

QA

(G)

3. Abstracts of Research Presentations

(1) Note on primitive ideals of enveloping algebras in prime characteristic---

by Yufeng Yao, Department of Mathematics, East China Normal University,

Shanghai 200241, China, E-mail: yaoyufeng139@sina.com

Abstract:

   Let $g$ be a restricted Lie algebra over  an algebraically closed field $F$ of characteristic $p>0$, $Z(g)$ the center of the universal enveloping algebra $U(g)$ of $g$. In this paper, we study primitive ideals of $U(g)$. The following results are included:

(i) The ideal of $U(g)$ generated by the central character ideal associated with any irreducible $g$-module has finite co-dimension in $U(g)$.

      Furthermore, the co-dimension is no less than $d(g)^2$, where $d(g)$ is the maximal dimension of irreducible $g$-modules

(ii) Each annihilator ideal of irreducible $g$-modules of maximal dimension is generated by

the corresponding central character ideal in $Z(g)$.

(iii) Each nonzero ideal in $U(g)$ intersects nontrivially with $Z(g)$.

(iv) Each $G$-stable ideal in $U(g)$ for $g=\Lie(G)$ contains non-zero fixed points under the action of $G$, where $G$ is a connected reductive group.

 

(2) Fermionic realization of two-parameter quantum affine algebra U_{r,s}(\hat{sl_n})---

by H. Zhang, Department of Mathematics, Shanghai University, China, E-mail:

Abstract:

  We construct all fundamental modules for the two parameter quantum affine algebra of typeA using a combinatorial model of Young diagrams.  In particular we also give a fermionic realization of the two-parameter quantum affine algebra.

Part II. Workshop

1. Schedule of Workshop

（Venue: Jiaoshi Zhi Jia, Lecture Hall 321 教师之家多功能厅321）

July 24-26

July 24, Saturday

July 25, Sunday

July 26, Monday

8:45-9:00

Registration

 

9:00-9:10

Welcome Speeches

9:10-10:00

Mathieu

Chari

Yi-Zhi Huang

10:00-10:30

Group Photo

Tea Break

10:30-11:20

N.Reshetikhin

Nanhua Xi

Naihuan Jing

11:30-12:30

Lunch

14:30-15:20

Helminck

Ruibin Zhang

Free

15:20-15:50

Tea Break

15:50-16:40

Quanshui Wu

Bin Shu

16:50-17:40

Jiao Zhang

Li Luo

18:00-19:30

Banquet

 

 

2．Abstracts of Workshop Talks

(1) Weyl modules for current algebras---
by V. Chari, Department of Mathematics, University of California, Riverside, USA, E-mail:

Abstract:
   We discuss some recent results on global and local Weyl modules of the Lie algebra of polynomial currents associated to a simple Lie algebra. We also study the space of homomorphisms between the global Weyl modules.

 

(2)  Cartan subspaces---

by Loek Helminck, Department of Mathematics, Campus Box 8205, North Carolina State University, Raleigh, NC 27695-8205, USA, E-mail:

Abstract: 

Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, $\sigma$ an involution of G defined over k, H a k-open subgroup of the fixed point group of  $\sigma$ and $G_k$ (resp. $H_k$) the set of k-rational points of G (resp. H).  The variety $G_k/H_k$ is a generalization of a real reductive symmetric spaces to arbitrary fields and is called a symmetric k-variety. For real and p-adic symmetric k-varieties the space $L^2(G_k/H_k)$ of square integrable functions decomposes into several series, one for each $H_k$-conjugacy class of Cartan subspaces of $ G_k /H_k$. In this talk we will discuss some recent results about  the $H_k$-conjugacy class of Cartan subspaces. 

 

 

(3) First order deformations and second cohomologies of grading-restricted vertex algebras---

by Yi-Zhi Huang, Department of Mathematics, University of Rutgers, USA,

E-mail:
Abstract:

   I will discuss my recent result showing that the set of equivalence classes of first order deformations of a grading-restricted vertex algebra corresponds bijectively with the a second cohomology of the algebra introduced recently by myself. I will also discuss some simple examples.

 

(4) Vertex operator realization of Jack polynomials---

by Naihuan Jing (NCSU and SCUT), E-mail:

Abstract:

   We give a new method to show that Jack polynomials of rectangular shapes are realized by lattice vertex operator of rank one, and we further generalize this result to marked rectangular shapes.

 

(5) Generalized Verma modules and irreducible modules for osp(n|2)---

by Li Luo, Department of Mathematics, East China Normal University,

Shanghai 200241, China, E-mail: lluo@math.ecnu.edu.cn

Abstract:

   The character formula of any finite dimensional irreducible module L_\lambda for Lie superalgebra osp(n|2) is determined in terms of characters of generalized Verma modules M_\nv. As a by-product, the decomposition of tensor module L_\lambda\otimes C^n|2, where C^n|2 is the natural representation, is obtained.

   The talk is organized as follows. Firstly, we present some background material on osp(n|2) and introduce an analogue of Kostant’s u-cohomology, by which we can construct the generalized Kazhdan-Lusztig polynomials for finite dimensional irreducible osp(n|2)-modules. Then the information of tensor modules M_\lambda\otimes C^n|2 and L_\lambda\otimes C^n|2 is studied. Finally the character formulae for all finite dimensional irreducible osp(n|2)-modules are obtained.

 

(6) On varieties determined by their jets---

by Olivier Mathieu (with R. Gurjar) , Universite de Lyon , E-mail:

Abstract:

A variety is not determined by its infinitisemal neighborhood along a subvariety. However, we prove it is the case when a reductive is acting, and the subvariety is the unique invariant orbit.  Then we derive some corollaries for the topology of geometric quotient.

 

(7) On a semiclassical limit in the quantum Knizhnik-Zamolodchikov equation---

by Nicolai Reshetikhin, Department of Mathematics, University of California, Berkeley, USA

E-mail: reshetik@math.berkeley.edu  

Abstract:

   The talk will focus on the semiclassical limit of quantum Knizhnik-Zamolodchikov equations. In this limit KZ equations degenerate into an isoconnection problem. The isoconnection problem is a difference analog of Schesinger isomonodromy deformation problem.

(8) Weyl Groups for Non-Classical Lie Algebras and Related Polynomial Invariants---

by Bin Shu, Department of Mathematics, East China Normal University, China,

E-mail: bshu@math.ecnu.edu.cn

Abstract: 

    Let (g, [p]) be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p > 0.  In this talk, we will first introduce the scheme of tori of maximal dimension, and generic tori of g. The scheme will give rise to a so-called Weyl group S(g) which is a finite group and coincides with the usual Weyl group of g in case g is a Lie algebra of classical type. We then present the group S(g) for Lie algebras of Cartan type and provide applications concerning polynomial invariants, and weight space decompositions. This is a joint work with Jean-Marie Bois and Rolf Farnsteiner.

 

(9) Calabi-Yau algberas under Hopf actions---

by Quanshui Wu, School of Mathematical Sciences, Fudan University, E-mail:

Abstract:

   The talk will be about Calabi-Yau Hopf algberas and Calabi-Yau graded algberas, especially when the smash product is still Calabi-Yau for a Calabi-Yau Hopf algbera acting on a Calabi-Yau graded algebra by using Van den Bergh Duality and Homological determinants. 

 

(10) Some quotient algebras of affine Hecke algebras---

by Nanhua Xi (Chinese Academy of Sciences), E-mail: nanhua@math.ac.cn
Abstract:

   An affine Hecke algebra has a big center. By modulo central characters one gets some finite dimensional quotient algebras. We will discuss a few particular quotient algebras.

 

(11) An elementary introduction to cyclic (co)homology of Hopf algebras ---

by Jiao Zhang, Department of Mathematics, East China Normal University,

Shanghai 200241, China, E-mail: zhangjiao@math.jussieu.fr

Abstract:

   In this talk, I will give the definition of cyclic (co)homology of Hopf algebras. Properties will be given to show its relation with Lie algebras and quantum groups. I will give concrete examples to illustrate how to calculate the cyclic (co)homology of a Hopf algebra if time is permitted.

 

(12) K-theory and cohomology of quantum homogeneous spaces---

by Ruibin Zhang, School of Mathematics and Statistics, University of Sydney, NSW 2006
Australia, E-mail: rzhang@maths.usyd.edu.au

Abstract:

Quantum analogues of equivariant K-theory and Dolbeault cohomology are studied for the quantum homogeneous space determined by any quantum parabolic subgroup of a quantum group $U_q(g)$. The higher equivariant K-groups are shown to coincide with those of the category of finite dimensional representations of the quantum Levi subgroup of the parabolic. The Dolbeault cohomology groups are determined in terms of representations of $U_q(g)$.

 

 

 

 

Information:  Minhang Campus Atlas

Computer Service:

 

Checking Emails at Computer Room 200A, in Math Building,

Opening Time: 8:30---11:00, 12:00---16:00, Each Monday, Wednesday, Friday.

 

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