Representations of current algebras

Author(s)

Henrique de Oliveira Rocha

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2020-12-02

Examining board members

Vyacheslav Futorny; Angelo Calil Bianchi; Lucas Henrique Calixto

• Advisor: Vyacheslav Futorny

Abstract

The first objective of this thesis is to study simple weight $\\mathcal G$-modules with finite dimensional weight spaces, where $\\mathcal G = \\mathfrak g \\otimes A$, $\\mathfrak g$ is a reductive Lie algebra with finite dimension and $A$ is an associative unital commutative algebra of finite type. In particular we show that such modules are given by evaluation modules and parabolic induced modules. The second objective is study commutative algebras of $U(\\mathfrak g _m(n))$, where $\\mathfrak g _m(n) = \\mathfrak g \\mathfrak l _n(\\mathbb C) \\otimes (\\mathbb C [t]/\\langle t^m angle )$. For $n \\leq 3$ or $m \\leq 2$, we will show the graded image of some algebraic independent generators of the center of $U(\\mathfrak g _m(n))$ form a regular sequence in the associated graded algebra. We will also prove that for $m>1$ the graded image of generators of the Gelfand-Tsetlin subalgebra of $U(\\mathfrak g _m(n))$ form a regular sequence if and only if $n=1$ or $n=2$. Finally, we will show the graded image of generators of the Bethe subalgebra form a regular sequence, if $n=2$ and $m \\geq 1$ or $n=3$ and $m=1$. These results implies the freeness of $U(\\mathfrak g _m(n)))$ over such commutative subalgebras and the existence of irreducible $\\mathfrak g _m(n)$-modules lifted by such algebras. (AU)

• FAPESP's process: 18/12382-9 - Representations of current Lie algebras
• Grantee: Henrique de Oliveira Rocha
• Support Opportunities: Scholarships in Brazil - Master