Cuspidal Representations of Lie Algebras

• Grant number: 20/14313-4
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: May 01, 2021
• End date: October 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Iryna Kashuba
• Grantee: Eduardo Monteiro Mendonça
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 18/23690-6 - Structures, representations, and applications of algebraic systems, AP.TEM
• Associated scholarship(s): 22/05915-6 - Cuspidal representations of Lie algebras and modules finitely generated over Cartan subalgebra, BE.EP.DR

Abstract

The main goal of this project is the study of combinatorial aspects of the representation theory of Lie Algebras, in particular their cuspidal representations. In a remarkable paper, O. Mathieu classified the irreducible weight modules by classifying cuspidal representations via coherents families. In his work, the author obtained combinatorial properties such as a formula for the degree of a cuspidal modules, which signs are involved. New combinatorial properties (without signs) are needed and expected. This will be our first goal. Another natural class of modules are the ones where the Cartan subalgebra h acts freely. J. Nilsson studied such class and described certain families of such irreducible modules by considering connections with coherent families of degree 1. His arguments hint that connection between the categories of weight modules and U(h)-free modules of finite type (in the Harish-Chandra sense) is much stronger and intend to explore in the case of type A. (AU)

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)

FUTORNY, VYACHESLAV; LOPES, SAMUEL A.; MENDONCA, EDUARDO M.. Representations of Smith algebras which are free over the Cartan subalgebra. Journal of Algebra, v. 655, p. 19-pg., 2024-06-10. (20/14313-4, 22/05915-6)