Mishchenko-Fomenko subalgebras of universal enveloping algebras of simple Lie algebras

Author(s)

Maria Clara Cardoso

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2019-08-02

Examining board members

Vyacheslav Futorny; Lucas Henrique Calixto; Plamen Emilov Kochloukov

• Advisor: Vyacheslav Futorny

Abstract

In this dissertation, we introduce the Mishchenko-Fomenko subalgebras. We show Vinberg\'s problem and the solution given by Feigin, Frenkel and Toledano-Laredo in Feigin, Frenkel and Toledano-Laredo (2010). We also show a solution for Lie algebras of type A found in Futorny and Molev (2015). We study the article Molev (2013) where generators for the Feigin-Frenkel center are shown for Lie algebras of type B, C and D. We introduce the Gelfand-Tsetlin subalgebras, which are subalgebras of the universal enveloping algebras of Lie algebras of type A. We show a definition of Gelfand-Tsetlin for Lie algebras of type C, introduced in Molev and Yakimova (2017). We exhibit the Gelfand-Tsetlin varieties related to $\\mathfrak_$ and $\\mathfrak_$. We prove that the Gelfand-Tsetlin variety for $\\mathfrak_$ is equidimensional of dimension 4 and we prove a new result about the equidimensionality of $\\mathfrak_$. (AU)

• FAPESP's process: 17/11050-0 - Mischenko-Fomenko subalgebras of universal enveloping algebras of simple Lie algebras
• Grantee: Maria Clara Cardoso
• Support Opportunities: Scholarships in Brazil - Master