Atlantic Association for Research in Mathematical Sciences
Memorial University of Newfoundland
Atlantic Algebra Centre
September 3 – September 10, 2008
Introduction to PI-algebras
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