Schedule 6/26 (Tue) 6/27 (Wed) 6/28 (Thu) 6/29 (Fri) 6/30 (Sat) 7/1 (Sum) 08:30¡X09:15 K. Harada A. V. Mikhalev E. Zelmanov Kenting Tour L. Carbone L. A. Bokut 09:45¡X10:30 S. J. Kang V. Latyshev L. W. Small S. J. Cheng M. A. Chebotar 11:00¡X11:45 C.-H. Liu G. Pilz K. I. Beidar M. Racine T.-K. Lee 14:00¡X14:45 C.-H. Lam L. Rowen A. Kemer T.-L. Wong 15:15¡X16:00 S.-Y. Pan V. Kharchenko A. A. Mikhalev J. Zhang 16:30¡X17:15 K. H. Lee L. Makar-Limanov Evening B.B.Q. Banquet

###### Abstracts

 June 26 08:30¡X09:15 K. Harada From PSp(4,3) to the Monster 09:45¡X10:30 S. J. Kang Crystal bases for quantum affine algebras and combinatorics of Young walls 11:00¡X11:45 C.-H. Liu Units of twisted group algebras 14:00¡X14:45 C.-H. Lam Codes, Moonshine VOA, and Monster 15:15¡X16:00 S.-Y. Pan Local theta correspondence and minimal K-types of positive depth 16:30¡X17:15 K.-H. Lee Hecke algebras, Specht modules and Gröbner-Shirshov bases June 27 08:30¡X09:15 A. V. Mikhalev Algebraic methods in measure theory 09:45¡X10:30 V. Latyshev An improved version of standard bases 11:00¡X11:45 G. Pilz Polynomials and Polynomial Functions 14:00¡X14:45 Recent progress in the structure of division algebras 15:15¡X16:00 V. Kharchenko Symmetric quantum Lie operations and differential calculi 16:30¡X17:15 June 28 08:30¡X09:15 E. Zelmanov Lie algebras over noncommutative rings and pseudoalgebras 09:45¡X10:30 L. W. Small From Jacobson rings to the Jacobson conjecture 11:00¡X11:45 K. Beidar On functional identities and their applications 14:00¡X14:45 A. Kemer On some problems in PI-Theory in characteristic p 15:15¡X16:00 A. A. Mikhalev Primitive elements of free algebras of Schreier varieties 16:30¡X17:15 June 29 Sight seeing tour June 30 08:30¡X09:15 L. Carbone Lattices in Kac-Moody groups 09:45¡X10:30 S.-J. Cheng Represention theory of the super W1+¡Û and its subalgebras 11:00¡X11:45 M. Racine Orders in simple algebras 14:00¡X14:45 T.-L. Wong On Amendariz rings 15:15¡X16:00 J. Zhang Homological properties of PI Hopf algebras 16:30¡X17:15 L. Makar-Limanov Cancellation for curves July 1 08:30¡X09:15 L. Bokut Gröbner-Shirshov bases for algebras and groups 09:45¡X10:30 M. Chebotar Derivations in prime rings 11:00¡X11:45 T.-K. Lee Skew derivations algebraic over prime rings 14:00¡X14:45 15:15¡X16:00 16:30¡X17:15

Abstracts

l         On functional identities and their applications

K. I. Beidar

Let R be a field, let S be a nonempty set, let £\ : S ¡÷ R be a map and let 0 ¡Ú f(x1, x2, ¡K, xn+1) £ R[x1, x2, ¡K, xn+1]. Recall that a function £p : Sn ¡÷ R is called algebraic if

f(x1£\, x2£\, ¡K, xn£\,£p(x1, x2, ¡K, xn)) = 0

for all x1, x2, ¡K, xn £ S. Now assume that degx_{n+1}(f) = 1. Then one can find £p from the above equation. Therefore one can find explicitly algebraic functions of degree 1. It happens that an analogous result is true (under mild assumptions on Im(£\) and £p) in the case when R is a noncommutative ring and f is a polynomial in noncommutative variables.

In the talk we shall discuss recent results on functional identities and their applications. Loosely speaking the theory of functional identities is the theory of ¡§algebraic functions of degree 1¡¨ in noncommutative rings. Functional identities were the main tool in solutions of long standing Herstein's problems on Lie and Jordan maps in noncommutative rings. They have applications to Lie-admissible algebras, graded polynomial identities and theory of linear preservers.

###### l         Gröbner-Shirshov bases for algebras and groups

L. A. Bokut

A.I. Shirshov (1962) invented a new general method of dealing with defining relations of any Lie algebra. Shirshov's method was explicitly formulated by the author in 1972 and then extended to associative algebras in 1976. Essentially, the same method for commutative algebras was discovered in a paper by H. Hironaka in 1964 and in a Thesis by B. Buchberger in 1965, under supervisions of W. Gröbner (published in 1970). Also G. Bergman in 1978 published some similar results for associative algebras. Later on the Sirshov method had been called the Gröbner-Shirshov bases method for non-commutative and Lie algebras.

Recently the Gröbner-Shirshov bases method had been extended to conformal algebras by the author, Y. Fong and W.-F. Ke. Also a general hypothesis was formulated for Gröbner-Shirshov bases of any Coxeter group joint with L.-S. Shiao. This hypothesis is proved for the Coxeter groups of the types An, Bn, Dn, as well as for some others cases.

l         Derivations in Prime Rings

Mikhail Chebotar

We shall briefly survey several topics on derivations in prime rings. In particular, solutions of Herstein's problems on Lie derivations (joint work with K. I. Beidar, M. Brešar and W. S. Martindale) and Lanski's problems on compositions of derivations (joint work with P.-H. Lee) will be presented.

l         Representation theory of the super W1+¡Û and its subalgebras

S.-J. Cheng

We study quasifinite representations of the Lie superalgebra of differential operators on the circle with N=1 extended symmetry, the so-called super W1+¡Û, from the perspective of Howe duality.  In particular we obtain a duality, in the sense of Howe, between super W1+¡Û and the general linear group GL(n).  We also exhibit a duality between a B-type subalgebra of the super W1+¡Û and Pin(2n).

###### l         From PSp(4,3) to the Monster

A (general) curve of degree 3 contains exactly 27 lines and the automorphism group of its configuration is PSp(4,3). Firstly a 3 x 3 matrix having (1,1) entry PSp(4,3) and (3,3) entry Monster is defined and its meaning is explored. Secondly the characterization of Sylow 2-subgroups of finite simple groups is discussed.

###### l         Crystal bases for quantum affine algebras and combinatorics of Young walls

Seok-Jin Kang

We give a realization of crystal bases for quantum affine algebras using some new combinatorial objects which we call the Young walls. The Young walls consist of colored blocks with various shapes that are built on the given ground-state wall and can be viewed as generalizations of Young diagrams. The rules for building Young walls and the action of Kashiwara operators are given explicitly in terms of combinatorics of Young walls. The characters of of basic representations can be computed easily by counting the number of colored blocks that have been added to the ground-state wall.

l         On some problems in PI-Theory in characteristic p

Alexander Kemer

We'll talk about some old and new results about the identities of PI-algebras over a field of characteristic p. Most of theresults are concerning the prime varieties.

###### l         Symmetric quantum Lie operations and differential calculi

V.K. Kharchenko

The notion of quantum Lie operation naturally appeared in line with the Friedrichs criteria for Lie polynomials (Journal of Algebra, 217, 188--228, 1999). We prove that the (n - 2)!-dimensional  space of generic quantum Lie operations has a basis of the symmetric ones. In the general case almost always a basis of symmetric operations exists. All exceptional cases are found. We propose a notion of a quantum universal enveloping algebra based on the quantum Lie operation concept. This enveloping algebra has PBW basis that admits a monomial crystallization. Every homogeneous character Hopf algebra over a field of zero characteristic is a quantum universal enveloping algebra of a suitable Lie algebra. We investigate in details a left covariant first order differential calculus that naturally arises on each skew primitively generated Hopf algebra with a week homogeneity condition. By means of the P. M. Cohn theory we show that the subalgebra of constants for the cover free differential algebra is a free algebra and an ad-invariant left coideal. We prove density and structural theorems for the operator algebra generated by partial derivatives. If the given algebra is finitely generated then every differential left ideal is generated by constants, a non-commutative Tailor series decomposition formula is valid, and the category of locally nilpotent modules over the operator algebra is semisimple with the only simple object that is isomorphic to the optimal algebra as a module. We find a necessary and sufficient condition for a 1-form to be a complete differential.

C.-H. Lam

###### l         An improved version of standard bases

Victor N. Latyshev

In order to extend the area of applications a comprehensive idea of standard bases is introduced in weak restrictions on simplifications. The well-known examples and new instances are described from a general point of view, and some recognizable properties of associative algebras are presented.

l         Hecke algebras, Specht modules and Gröbner-Shirshov bases

K. H. Lee

I will talk about Grobner-Shirshov bases for Specht modules of Hecke algebras of type A. The structure of Specht modules will be explained using the combinatorics of Young walls. As an application, an algorithm of computing Gram matrices will be discussed.

l         Skew derivations algebraic over prime rings

T.-K. Lee

Let R be a prime ring with extended centroid C and left Martindale quotient ring R{\cal F}. In this paper we first give a characterization of £m-derivations which are left algebraic over R{\cal F} modulo finite-dimensional subspaces of R{\cal F}. Applying this characterization we prove some results concerning £m-derivations. This is a joint paper with Professor Chen-Lian Chuang.

l         Units of twisted group algebras

Chia-Hsin Liu

Let U be the group of units of a twisted group algebra. We are interested in the following two problems:

1.      When does U satisfy a group identity?

2.      When does U contain a free subgroup of rank two?

###### l         Cancellation for curves

L. Makar-Limanov

Here is the cancellation theorem of Abhyankar-Eakin-Heinzer, see S. Abhyankar, P. Eakin, W. Heinzer, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), 310--342.

Let £F1 and £F2 be two curves and O(£F1), O(£F2) be the rings of regular functions on £F1 and £F2. If O(£F1)[x1, ..., xn] \IsomorphicTo O(£F2)[x1, ..., xn], then O(£F1) \IsomorphicTo O(£F2).

In the talk a new proof of this theorem in the case of characteristic zero which is based on the computations of invariant \AK and some conjectures will be discussed.

l         Primitive Elements of Free Algebras of Schreier Varieties

Alexander A. Mikhalev

We expose results on primitive elements of free algebras of main types of Schreier varieties of algebras. A variety of linear algebras over a field is Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras. A system of elements of a free algebra is primitive if it is a subset of some set of free generators of this algebra. We consider free non-associative algebras, free commutative and anti-commutative non-associative algebras, free Lie algebras and superalgebras, free Lie p-algebras and p-superalgebras. We present matrix criteria for systems of elements to be primitive. It gives a possibility to obtain algorithms to recognize primitive systems of elements. Primitive elements distinguish automorphisms: endomorphisms sending primitive elements to primitive elements are automorphisms. Finally, we consider stable and dense properties for primitive elements.

###### l         Algebraic methods in measure theory

Alexander V. Mikhalev

The following new algebraic lines in measure theory (developed recently by A. V. Mikhalev and V. K. Zakharov) are under consideration:

2.      Rings and modules related to measure theory.

3.      Thin linear functionals and metasemicontinuous functions.

4.      Integral representation for Radon (bi)measures over an arbitrary Hausdorff space (a solution of Riesz-Radon problem)

l         Lcal theta correspondence and minimal K-types of positive depth

S.-Y. Pan

In this talk, we want to discuss the relation between the minimal K-types of the irreducible admissible representations paired by theta correspondence. In particular, we show that the minimal K-types are paired by orbit correspondence when the depth is positive.

###### l         Polynomials and polynomial functions

G. Pilz

Polynomials over rings are pretty well-known. What about polynomials over general algebraic structures? Even if we stay in the case of rings: what is the relation to polynomial functions, can we always ¡§compare coefficients¡¨? What you always wanted to know abut commuting and/or non-commuting variables, but never dared to ask. Why are polynomials so useful for describing ideals in Omega-groups? For the description of the structure of universal algebras via commutators? Every polynomial map is ¡§compatible¡¨: it maps congruent elements into congruent ones. When is every map on an algebra a polynomial map (polynomial completeness)? When is every compatible function a polynomial one (affine completeness)?

###### l         Recent progress in the structure of division algebras

L. Rowen

The major question in the structure theory of division algebras may be to express a given F-division algebra D in terms of cyclic algebras. Although the Merkurjev-Suslin Theorem says that (assuming F has enough roots of 1) some matrix ring Mt(D) is isomorphic to a tensor product of cyclic algebras, no bound is given on t. Indeed it has long been known t = 1 for D of degree 2, 3, and 6, and t > 1 if deg (D) is divisible by a square number > 1. On the other hand, it is unknown whether or not t = 1 for deg(D) square-free. Our main result (jointly with Lorenz, Reichstein, and Saltman) is that if deg(D) = 4 then t = 2, i.e. M2(D) is similar to a tensor product of two cyclic algebras, of respective degrees 4 and 2. The proof relies on earlier structural results of Albert, Amitsur-Saltman, and Tignol. The theorem can be seen in a broader context using the essential dimension, and I will describe what is known about the essential dimension of central simple algebras.

l         From Jacobson Rings to the Jacobson Conjecture

Lance W. Small

I'll give a survey of the results and background of the Jacobson Conjecture and, then, relate it recent developments in affine algebras.

l         Applications of combinatorial algebra to cryptography

V. Shpilrain

This talk will focus on recent applications of algebra to cryptography. This area is becoming increasingly popular because most of the presently existing computer security systems

are becoming outdated due to a dramatic increase in the speed of computation offered by modern computers. There is therefore a high demand for brand new ideas in this area. We are going to survey some of them.

l         On Armendariz rings

Tsai-Lien Wong

A ring R is called Armendariz if whenever polynomials f(X) = a0 + a1X + ¡K + amXm and g(X) = b0 + b1X + ¡K+ bnXn in R[X] satisfy f(X)g(X) = 0, then aibj = 0 for all i, j. A ring R is called reduced if it has no nonzero nilpotent elements. The study of Armendariz rings was initiated by Armendariz, Rege and Chhawchharia.

In this talk we are concerned with the connections among (weak) Armendariz rings, reduced rings and semiprime right Goldie rings. We construct certain Armendariz rings and prove that a semiprime right Goldie ring is weak Armendariz if and only if it is a reduced ring.

###### l         Lie algebras over noncommutative rings and pseudoalgebras

E. Zelmanov

l         Homological properties of PI Hopf algebras

J. Zhang

Quantum groups at the roots of unity form a family of noetherian PI Hopf algebra. This family of algebras have nice homological properties such as the Auslander-Buchsbaum formula, Bass theorem, and the Gorenstein property. The talk will review some results of Brown and Goodearl and some recent work of Wu by using the noncommutative version of dualizing complexes introduced by Yekutieli.