Noncommutative Algebra

• Grant number: 23/01710-3
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: April 01, 2023
• End date: March 31, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Plamen Emilov Kochloukov
• Grantee: Marcos Emanuel Veríssimo do Nascimento
• 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 shall study the fundaments of the theory of noncommutative rings. We shall cover division rings and the Frobenius theorem about the quaternion algebra. The Wedderburn theorem concerning the commutativity of finite division rings will be covered as well. We shall see also the notion of a central simple algebra. The Skolem-Noether theorem about the automorphisms will be a central piece in our studies. We will pass then to the Wedderburn-Artin theorem concerning the structure of algebras with a finiteness condition: first in the finite dimensional case and then in the artinian case. Then we pass to the notion of a tensor product from the point of view of central simple algebras. We shall define the Brauer group, and we will compute it in some classical situations. Finally we shall prove that the Brauer group of the rational numbers is infinite by means of constructing infinitely many division algebras of dimension 4 over the rationals which are not isomorphic.