|
Schedule
Titles and Abstracts:
Zongzhu Lin
Lecture 1. Group schemes and and representations of group schemes.
Lecture 2. Cohomology theory of group schemes, finite generation theorem for finite group schemes.
Lecture 3. Cohomological varieties and support varieties for infinitesimal group schemes, finite group schemes and p-points.
Lecture 4. Ranks varieties and Parshall Friedlander description of the support varieties for restricted Lie algebras, relating support varieties for restricted Lie algebras and finite groups of Lie types, Friedlander's Weil extension approach.
References:
1) J.C. Jantzen: Representations of algebraic groups,
2) D. Benson, Representations and cohomology vol 1-2
3) E. Friedlander and Sulin, cohomology of finite group schemes,Inventiones math
4) Friedlander and Parshall, support varieties of restricted Lie algebras. Inventiones,
5) Friedlander and Pevtzova, p-points for finite schemes groups
Jingsong Huang
Title:Dirac Operators in Representation Theory
Abstract: The content of the course is based on the book which Pavle Pandzic and I wrote and was published in Birkhauser book series Mathematics: Theory & Applications in 2006.We will also include the most recent developments after
the book was ublished. The aim of the course is to show
that using Dirac operators as a unifying theme, many of the most
important results in representation theory fit together when viewed
from this perspective.
Title of the talk for the conference:
Dirac cohomology of Harish-Chandra modules
Abstract: Dirac operators are widely used in physics and differential
geometry as well as geometric construction of group representations.
The related concept Dirac cohomology is a generalization of the
idea in index theory
to representation theory. We demonstrate how to calculate the Dirac
cohomology
for some of the most interesting representations of reductive Lie groups and to obtain
some applications.
Yun Gao
Lecture 1: Affine Kac-Moody Lie algebras
Lecture 2: Toroidal Lie algebras
Lecture 3: Representations via vertex operators
Lecture 4: Representations via bosons and fermions
References:
(1)V.Kac, Infinite dimensional Lie algebras. 1990
(2)R.Moody,A.Pianzola, Lie algebras with triangular decomposition. 1995
(3)I.Frenkel, J.Lepowsky,A.Meurman, Vertex Operator Algebras and the Monster. 1989
Bangming Deng:
Title: Ringel-Hall algebra approach to quantum groups
In this series of lectures we give a brief introduction on Ringel-Hall algebra approach to quantum groups. Some interactions between representation theory of algebras and quantum groups will also be discussed. This will be given in the following four talks:
Lecture 1: Ringel-Hall algebras and quantum Serre relations
Lecture 2: Green’s comultiplication and realization of positive parts of quantum enveloping algebras
Lecture 3: Structure of Ringel-Hall algebras and a generalization of Kac theorem
Lecture 4: Bases of quantum enveloping algebras of finite type
References:
[1] B. Deng, J. Du, B. Parshall and J. Wang, Finite dimensional algebras and quantum groups, Mathematical Surveys and Monographs Volume 150, Amer. Math. Soc., Providence 2008.
[2] B. Deng and J. Xiao, A new appraoch to Kac's theorem on representations of valued quivers, Math. Z. 245(2003), 183-199.
[3] J. A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120(1995), 361-377.
[4] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc., 3(1990), 447-498.
[5] G. Lusztig, Introduction to quantum groups, Progress in Math. 110, Birkh\"auser, 1993.
[6] C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101(1990), 583-592.
[7] C. M. Ringel, PBW-bases of quantum groups, J. reine angew. Math. 470(1996), 51-88.
[8] B. Sevenhant and M. Van den Bergh, A relation between a conjecture of Kac and the structure of the Hall algebra, J. Pure and Appl. Algebra 160(2001), 319-332
Nicolas Monod
Title:Unitarisability of representations
Following Sz.-Nagy's characterisation of operators conjugated to a unitary, Dixmier and Day proved in 1950 that every uniformly bounded representation of an amenable group is unitarisable. Dixmier asked whether the converse holds; this question remains open as of today. In 1955, Ehrenpreis and Mauntner discovered that this is not the case for SL(2,R). In the 1980's, several explicit descriptions of non-unitarisable representations of free group were given.
In these lectures, we shall review the above material and present new results obtained with I. Epstein and N. Ozawa. The starting idea is to use ergodic theory and random graphs. We shall explain ideas originating with a 1847 paper by Kirchhoff on electrical networks. As one application, we will explain why certain Burnside groups are non-unitarisable.
Kaiming Zhao：
Title: First cohomology group and automorphism group of nilradicals in Kac-Moody algebras
Lecture 1: Finite-dimensional simple Lie algebras
Lecture 2: Kac-Moody algebras
Lecture 3: Moody’s Conjecture
Abstract: I will start this series of talks from scratch, from understanding classification of Finite-dimensional simple Lie algebras, introducing a classification of their irreducible weight modules obtained recently by Mathieu, recalling some results of Kac-Moody algebras, to answering Moody’s conjecture.
In 1980, R. V. Moody posed a conjecture in his paper [M] about first cohomology group of nilradicals of the standard Borel subalgebras in Kac-Moody algebras. In case of finite type the first cohomology group were given in [Ko] and [LL].
In the case of affine type type the first cohomology group was obtained in [F].
I conclude my talks with answering the remaining case of Moody’s Conjecture and giving the automorphism group of nilradicals in Kac-Moody algebras.
Bin Shu:
Title: Weyl Groups for Restricted Lie Algebras and Related Invariants
Abstract: Let (g, [p]) be a restricted Lie algebra, defined over
an algebraically closed field k of characteristic p > 0. The scheme
of tori of maximal dimension gives rise to a finite group S(g) that
coincides with the Weyl group of g in case g is a Lie algebra of
classical type. In this talk, we compute the group S(g) for Lie
algebras of Cartan type and extend results by Premet concerning
invariants of Jacobson-Witt algebras to arbitrary restricted simple
Lie algebras. This is a joint work with R. Farnsteiner.
References:
[K] V. G. Kac, Infinite dimensional Lie algebras, 3rd ed., Cambridge Univ. Press, 1983.
[Ko] B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math., 74(1961), no.2, 329-387.
[LL] G. F. Leger, E. M. Luks, Cohomology theorems for Borel-like solvable Lie algebras in arbitrary characteristic, Trans. Amer. Math. Soc., 195(1974), 305-316.
[F] A. Fialowski, On the cohomology of infinite dimensional
nilpotent Lie algebras, Adv. Math., 97(1993), no.2, 267-277.
[Ma] O. Mathieu, Classification of irreducible weight modules. Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 537--592.
[M] R. V. Moody, Isomorphisms of upper Cartan-matrix Lie algebras,
Proc. London Math. Soc., 40(1980), no.3, 430-442.
[MZ] J. Morita, K. Zhao, Automorphisms and derivations
of Borel subalgebras and their nilradicals in Kac-Moody algebras,
arXiv:0806.4922.
|
Center of Mathematical Sciences, Zhejiang University