Graded tensor product and regular algebras

• Grant number: 23/01673-0
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: June 01, 2023
• End date: February 28, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Plamen Emilov Kochloukov
• Grantee: Kauê Orlando Pereira
• 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
• Associated scholarship(s): 25/03763-2 - Infinite-dimensional regular gradings and regular gradings that are homogeneously graded Morita equivalent., BE.EP.DR

Abstract

In this project for PhD research we shall study topics in the theory of algebras with polynomial identities. The notion of a regular algebra was introduced by Regev and Seeman, and afterwards was studied by Bahturin and Regev, and various other researchers. The decomposition of a regular algebra into a (finite) direct sum of vector subspaces satisfying certain properties (see below) can often be considered as a grading on the algebra by a finite abelian group. Regular algebras appear in a natural way in the study of graded tensor products. More generally, when the regular decomposition is a group grading, one studies twisted tensor products where the twist is given by a skew-symmetric bicharacter (or, equivalently, by a 2-cocycle on the grading group). We shall study the multilinear identities in regular algebras and their twisted tensor products. Our main goal will be the study and solution of two conjectures proposed by Bahturin and Regev. The first of these considers the minimality of a regular decomposition, and relates it to tye invertibility of its associated matrix. The second conjecture states that the quantity of summands in a regular minimal decomposition of an algebra as well as the determinant of the corresponding matrix, are invariants of the algebra, and do not depend on the choice of the decomposition. We shall also be looking for descriptions of this associated matrix.

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

CENTRONE, LUCIO; KOSHLUKOV, PLAMEN; PEREIRA, KAUE. A negative answer to a Bahturin-Regev conjecture about regular algebras in positive characteristic. Journal of Pure and Applied Algebra, v. 229, n. 6, p. 20-pg., 2025-06-01. (23/01673-0, 18/23690-6)