Gelfand-Tsetlin modules for Lie algebras

• Grant number: 12/23450-9
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: April 01, 2013
• End date: September 30, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Vyacheslav Futorny
• Grantee: Luis Enrique Ramírez
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

The main goal of this project is to describe the irreducible Gelfand-Tsetlin modules over the complex Lie algebra sl(n). This problem is of huge importance for the representation theory of Lie algebras, where the complete classification of irreducible modules is known only when n = 2. To reach our objective we plan to use the results obtained by the candidate for the case of sl(3) during the development of his PhD thesis. Further, in [1] and [10] it was shown how to describe the category O using blocks parametrized by central characters, such that each block is equivalent to the category of A-modules for some finite-dimensional associative algebra A. The same method could be applied to the category of Gelfand-Tsetlin modules. In this direction we intend to obtain a description of the blocks that allows us: to study the internal structure of the blocks using as a main tool the localization functor, to relate the blocks (in some cases give equivalence of the categories) using the translation functor and, finally, to use the techniques of representation theory of finite dimensional algebras to describe the quivers associated with the different blocks. (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; GRANTCHAROV, DIMITAR; RAMIREZ, LUIS ENRIQUE. Singular Gelfand-Tsetlin modules of gl(n). ADVANCES IN MATHEMATICS, v. 290, p. 453-482, FEB 26 2016. (11/21621-8, 14/09310-5, 12/23450-9)

FUTORNY, VYACHESLAV; GRANTCHAROV, DIMITAR; RAMIREZ, LUIS ENRIQUE. Irreducible Generic Gelfand-Tsetlin Modules of gl(n). Symmetry Integrability and Geometry-Methods and Applications, v. 11, 2015. (11/21621-8, 14/09310-5, 12/23450-9)