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Cuspidal Representations of Lie Algebras

Grant number: 20/14313-4
Support Opportunities:Scholarships in Brazil - Doctorate
Effective date (Start): May 01, 2021
Effective date (End): 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)

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