Infinite-dimensional regular gradings and regular gradings that are homogeneously graded Morita equivalent.

• Grant number: 25/03763-2
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• Start date: September 01, 2025
• End date: August 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Plamen Emilov Kochloukov
• Grantee: Kauê Orlando Pereira
• Supervisor: Lucio Centrone
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: Università degli Studi di Bari - Aldo Moro, Italy
• Associated to the scholarship: 23/01673-0 - Graded tensor product and regular algebras, BP.DR

Abstract

The main goal of this project is to study the algebraic structure of regular gradings that have a minimal regular decomposition. The first part aims to provide a complete description of such gradings that are infinite-dimensional, since the structure of finite-dimensional regular gradings with minimal regular decomposition has already been obtained in recent results. The hypothesis of infinite dimension and the minimal regular decomposition lead us to suspect that there is an important relationship between the Grassmann algebra $E$ and such algebras, which may be fundamental in describing the elements of the Jacobson radical of these algebras, directly influencing their regular decomposition matrix. The second proposed problem aims to investigate under which conditions two regular gradings are Morita equivalent (in the graded context) and how the minimality of the regular decomposition is treated in this context. We believe that this last problem broadens the horizons of the study of regular gradings and, furthermore, provides a good connection with noncommutative geometry, enabling a branch of study of these algebras from a geometric perspective.