Álgebras e superálgebras fundamentais e a classificação de variedades com multiplicidades limitadas por um

Abstract

Let A be an algebra over an algebraically closed field of characteristic zero. The representation theory of the symmetric group Sn plays an important role in the study of the polynomial identities of an algebra A. In this context, one associates to A a sequence of Sn-characters {χn(A)}n≥1, called cocharacters of A, and a central problem is to determine the decomposition of each χn(A) into irreducible characters. In this thesis, we classify, up to PI-equivalence, the fundamental algebras whose cocharacter sequences have multiplicities bounded by one. As a consequence, we obtain a complete classification, up to PI-equivalence, of all algebras whose cocharacters have multiplicities bounded by one. Furthermore, we extend the presented study of fundamental algebras with multiplicities bounded by one to the context of superalgebras and show that, up to isomorphism, Dgr is the only Z2-fundamental superalgebra with Z2-exponent equal to 2 and with Z2-multiplicities bounded by one.