Polynomial Identities in Matricial Algebras

• Grant number: 24/18779-9
• Support Opportunities: Scholarships in Brazil - Master
• Start date: March 01, 2025
• End date: February 28, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Plamen Emilov Kochloukov
• Grantee: Gabriel Cruz Vitale Torkomian
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

In this master's project we shall study classical topics in the theory of algebras with polynomial identities, namely the identities satisfied by matrix algebras. We shall begin with a brief introduction to the area: polynomial identities, PI algebras, varieties of algebras, T-ideals, the linearization process, homogeneous and multilinear structure of relatively free algebras, the theory of representations of the symmetric group and the general linear group. In this part, the material studied during the scientific initiation, with a grant from this Foundation, will be welcome, especially the theory of group representations and the structure of rings and algebras. We shall continue with the applications of the theory of representations of the symmetric group to the theory of PI algebras. We will see the polynomial identities of the Grassmann algebra, as well as upper triangular matrices. As an application we shall cover Razmyslov's fundamental results on the identities of the Lie algebra sl_2(K), the weak identities of the pair (M_2(K), sl_2(K)), and the matrix algebra M_2(K ), when K is a field of characteristic 0.The identities satisfied by the matrices of order n>2, have not been described yet. However, a theorem, proved simultaneously (and independently) by Procesi and Razmyslov, gave a complete description of the trace identities in M_n(K). Procesi's approach involves notions of great importance in PI theory: generic matrices and the theory of invariants of classical groups. We shall follow Procesi's ideas to unravel the proof of this important result. To do so, we have to study fundamental topics concerning the invariants of classical groups, which is also of independent interest. Razmyslov and Procesi's theorem has an unexpected and very interesting application: the Nagata, Higman, Dubnov and Ivanov theorem. This theorem says that if an associative algebra, in characteristic 0, satisfies the identity x^n=0, for some n, then it is nilpotent. The nilpotent index is not known if n>4, but the upper bound best known is obtained using trace identities.At the end of the project we intend to study the identities satisfied by the matrix algebra of order 2, when the base field is infinite and of characteristic greater than 2. We will need to study the weak identities, as well as the identities of sl_2(K), before proceeding with those of M_2(K). We hope that at the end of the master's degree the student will have the knowledge to start a doctorate in algebra, more specifically in combinatorial theory of rings and algebras.