INTERNATIONAL CONFERENCE ON ALGEBRA AND ITS APPLICATIONS
Southern Taiwan University of Technology, Tainan, Taiwan
December 19–24, 2005
Abstracts
Matej Brešar (Slovenia) Title:Commuting maps
Abstract. A mapf from a ringRinto itself is said to be commuting iff .x/commutes withx for everyx inR. We shall survey the development of the theory of commuting maps and their applications.
Shun-Jen Cheng (Taiwan)
Title:Canonical Basis inm.V/˝m.V/
Abstract: LetV is the naturalUq.gl.1//-module. In this talk we give combinatorial formulas for the canonical and dual canonical bases in the Fock space m.V/˝ m.V/. These basis elements admit representation-theoretical interpretation as Kazhdan-Lusztig polynomials in a certain parabolic category of gl.mCn/-modules.
This is a joint work with W. Wang and R.B. Zhang.
Alexandre Kirillov (USA)
Title:Matrices with non-commuting entries and family algebras
Abstract. Consider the matrixM D kEijkwith entries inU.gl.n//. It satisfies the analogue of the Cayley-Hamilton identity:
MN D XN
kD1
akMN k
whereak are some elements ofZ.g/.
For a general matrixAwith entries from a non-commutative algebraA there is no reason to have something like this. There is however a case when the powers ofAare linearly dependent
over the centerZofA. It happens, whenAis similar to a matrix with elements fromZ. We show that it is exactly the case when we consider the matrices which belong to a so-called quantum family algebra:
Q.g/D.U.g/˝End.V//G
Recall that a generic matrix over a commutative field can be reduced to the so-called second normal form:
AD
0 1 0 : : : 00 0 1 : : : 0
0 0 0 : : : 0
0 0 0 : : : 1
an an 1 an 2 : : : a1
˘
:
This property also generalizes to elements of a quantum family algebra.
Recently in the joint paper with L. Rybnikov we introduced the notion of odd family algebras.
I hope that the study of family algebras will help to understand the structure of U.g/ for semisimple Lie algebrasgand some other non-commutative objects.
Ching-Hung Lam
Title:Ising vectors and McKay’sE8observation on the Monster simple group
Abstract. In this talk, we shall discuss certain weight two vectors (called Ising vectors) in the VOAVp2E8 andVC. We shall discuss McKay’sE8-observation on the Monster simple group using these Ising vectors.
Leonid G. Makar-Limanov (USA)
Title:Ten years of AK-invariant: successes and disappointments
Abstract. AK invariant was introduced about ten years ago in order to prove that certain three- folds are exotic structures: it means that they are very similar to the usual three-dimensional space and that difference is too subtle to notice it by looking at small neighborhoods of points.
The invariant turned out to be useful in other settings as well and allowed to obtain some new results and new proofs of old results which are by far more simple then the proofs known before.
But unfortunately we still cannot answer a lot of natural questions about the behavior of the invariant.
Edmund R. Puczyłowski (Poland) Title:Around Koethe’s nil ideal problem
Abstract.In 1930 Köthe asked whetherevery left ideal of an arbitrary associative ring contained
in the nil radical of the ring? Though this question looks very elementary and was raised more than seventy years ago it is still open. It is calledKoethe’s nil ideal problemand is one of the most famous open problems in ring theory. Attempts to solve it led to many interesting, deep and sometimes surprising results. There are also many related open problem. The aim of the talk is to present several such problems as well as some new results obtained in the area. In particular it will be shown that Koethe’s problem is equivalent to a problem raised in 1969 by Andrunakievich.
Weiqiang Wang (USA)
Title:The Bloch-Okounkov correlation functions and generalizations
Abstract: Bloch and Okounkov introduced a certain n-point correlation functions on a Fock space several years ago and found a remarkable closed formula in terms of theta functions. These n-point functions admit geometric interpretations in terms of Gromov-Witten theory and Hilbert schemes of points. After reviewing some of these results, I will formulate several natural general- izations of thesen-point functions (with motivations from twisted Fock spaces, vertex operators, and Macdonald polynomials, etc) and present closed formulas in some cases.
This is a joint work with Shun-Jen Cheng.
Eugene Xia (Taiwan)
Title:The mapping class group action on the moduli of representations
Abstract.LetS be an oriented surface of genusgwith boundaryC andKa compact Lie group.
ThenKacts by conjugation on HomC.1.S/;K/, the space of representations of the fundamen- tal group1.S/intoK. The resulting quotient is the moduli of representations HomC.1.S/;K/=K.
This moduli space has a canonical topology and measure. The mapping class group of S acts on 1.S/, hence, acts on HomC.1.X/;K/=K and one may study both the topological and mea- sure dynamics of this action. This talk will focus on the measure dynamics of this action. We shall also outline a few results in topological dynamics if time permits.
Jing Yu (Taiwan)
Title:Algebraic relations among special values in characteristicp
Abstract. The goal of transcendence theory is to determine completely the algebraic relations among special values which naturally arise from arithmetico-geometric structures. In the positive characteristic world, the arithmetico-geometric structures are those from global function fields.
Here more advances can be made. We will report on two cases where the algebraic relations are completely determined recently, one is certain zeta values, the other is on Gamma values. We will also report on the algebraic tools developed so far for the solution of our problems.
Jiu-Kang Yu (USA)
Title:Quasi-reductive group schemes
Abstract: We study group schemes over a discrete valuation ring with reductive groups as generic fibers and reduced special fibers, and applications to the construction of Chevalley group by Mirkovic and Vilonen.
This is a joint work with G. Prasad.
Ruibin Zhang (Australia)
Title:A unified treatment of Schur-Weyl dualities for Lie algebras and for quantum groups Abstract. Let G be the complex Lie group GLn, On, Sp2n, or G2, and let V be the natural G-module if G is a classical group, or the 7-dimensional irreducible module of G2. We give a unified description of the endomorphism algebra EndG.V˝k/for any non-negative integer k by using the associative algebra Tk of “infinitesimal braids" due to Kohno and others. More specifically, there is a representation of Tk such that EndG.V˝k/ D .Tk/. The algebra Tk and the construction of the representation will be described, and a proof of the claimed equality will be given. The generalization of the result to quantum groups will also be presented.
This is joint work with G. I. Lehrer.
