Trace identities and invariant theory

Author(s)

Gabriel Guzatti Vilas Boas

• Document type: Master's Dissertation
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2025-02-26

Examining board members

Plamen Emilov Kochloukov; Claudemir Fideles Bezerra Junior; Antonio Ioppolo

• Advisor: Plamen Emilov Kochloukov

Abstract

In this work we present classical topics in the theory of algebras with polynomial identities. We begin by defining polynomial identities, PI algebras, varieties of algebras, and T-ideals, later on we show the process of multilinearization and its consequences. Afterwards, we focus on the study of M_n(K), the algebra of n x n matrices. We see the Amitsur-Levitski theorem, which shows that the Standard polynomial St_{2n} is the minimal-degree polynomial identity for M_n(K). We also study central simple algebras, and prove the Skolem-Noether theorem. We deduce, as a direct corollary, that the automorphisms of M_n(K) that fix the base field K are inner. Next, we study the action of the general linear group on m-tuples of matrices by conjugation. Following the fundamental work of Procesi, we study the description of the invariants of this action, leading to the first and second fundamental theorems of the invariant theory of matrices. As a consequence, we obtain a description of the trace identities satisfied by the algebra of n x n matrices. As an application, following Razmyslov's work, we derive an upper bound for the nilpotency index d(n) in the Nagata-Higman theorem. Additionally, we deduce the lower bound established by Kuzmin (AU)

• FAPESP's process: 23/01635-1 - Trace identities and Invariant theory
• Grantee: Gabriel Guzatti Vilas Boas
• Support Opportunities: Scholarships in Brazil - Master