Atlantic Association for Research in Mathematical Sciences

Memorial University of Newfoundland

Atlantic Algebra Centre

*September 3 – September 10, 2008*

Mini Course

**Introduction to PI-algebras**

**Vesselin Drensky**

Associative algebras with polynomial identities (or PI-algebras) are an important class of algebras which enjoy many of the properties of finite dimensional algebras and commutative algebras. The mini-course is devoted to the following topics.

**Polynomial Identities and Central Polynomials of Matrices**

Matrix algebras are among the most important and attractive to study noncommutative objects in Ring Theory, with numerous applications in mathematics and other branches of science. We present two of the main theorems on polynomial identities of matrices: the Amitsur-Levitzki theorem that the

*n*-by-*n*matrices satisfy the standard identity of degree 2n and the existence of central polynomials.**Structure Theory of PI-Algebras**

We consider several theorems which are among the cornerstones of the theory of PI-algebras: the Kaplansky theorem about primitive PI-algebras, the Levitzki theorem that primitive PI-algebras have no nil-ideals and the theorem of Posner, a localization theorem, which generalizes the fact that a commutative domain has a quotient field.

**Nagata-Higman Theorem and the Shirshov Height Theorem**

These two theorems (stated and proved in a purely combinatorial way) have many applications: the positive solution for PI-algebras of the Kurosh problem on the finite-dimensionality of finitely generated algebraic algebras, the theorem of Razmyslov-Kemer-Braun on the nilpotency of the radical of a finitely generated PI-algebra, the theorem of Berele that finitely generated PI-algebras have finite Gelfand-Kirillov dimension.

**Schedule of Lectures**

**Lecture 1**– Friday, September 5, 2 – 2:50 p.m.,

**Lecture 2**– Monday, September 8, 3 – 3:50 p.m.,

**Lecture**3 – Tuesday, September 9, 2 – 2:50 p.m

**., Lecture 4**– Wednesday, September 10, 3 – 3:50 p.m.,

*Atlantic Algebra Centre room HH-2010*