Polynomial identities and their numerical invariants

• Grant number: 20/16595-7
• Support Opportunities: Scholarships in Brazil - Master
• Start date: December 01, 2021
• End date: February 28, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Plamen Emilov Kochloukov
• Grantee: Kauê Orlando Pereira
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 18/23690-6 - Structures, representations, and applications of algebraic systems, AP.TEM

Abstract

We will study some topics in the Theory of algebras satisfying polynomial identities in this proposal for a Master Degree. We will approach the basics of the theory, namely polynomial identities, PI algebras, varieties of algebras, T-ideals, the linearization process, the homogeneous and the multilinear structures of relatively free algebras. To this end we shall need some facts from Ring theory such that the theorems due to Wedderburn and Artin, Wedderburn and Malcev, and also the theory of representations of the symmetric and of the general linear group. We shall cover the topics in the depth needed for our project. Moreover we shall study the notions of a codimension, codimension growth. We shall introduce the notion of the PI exponent of an algebra. We shall see the celebrated theorem due to A. Regev about the exponential growth of the codimensions, and we shall see how to apply it in order to prove another theorem due to Regev: that the tensor product of two PI algebras is once again a PI algebra. Afterwards we shall cover the theorem due to Giambruno and Zaicev that shows the PI exponent of an associative PI algebra in characteristic 0 always exists and is a nonnegative integer. Finally we shall describe the PI algebras whose codimensions are of polynomial growth.

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

PEREIRA, Kauê Orlando. Polynomial identities and their numerical invariants. 2023. Master's Dissertation - Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica Campinas, SP.