Problema do isomorfismo para álgebras envolventes universais de álgebra de lie

Abstract

Let L and H be two Lie algebras and UL and UH be their respective universal enveloping algebras. The isomorphism problem for universal enveloping algebras of Lie algebras asks us to determine under what conditions the isomorphism between UL and UH implies the isomorphism between L and H. In this dissertation, we will investigate this problem in the context of threedimensional simple Lie algebras and finite-dimensional nilpotent Lie algebras, based on the articles of Malcolmson [Mal92] and of Riley and Usefi [RU07]. In order to present a detailed proof of the main result of Malcolmson [Mal92], we will describe the class of quaternion algebras, based on the article [Lew06] of David W. Lewis and on masters dissertation [Sha08] of Zi Yang Sham. We will characterize the quaternion algebras over specific fields, in order to obtain a characterization of three-dimensional simple Lie algebras. Moreover, we will show that isomorphism between universal enveloping algebras of two three-dimensional simple Lie algebras occurs if and only if the Lie algebras are isomorphic. We will also show, based on the article [RU07], that the isomorphism type of the universal enveloping algebra of a finite dimensional Lie algebra determines whether or not the Lie algebra is nilpotent and, in case it is, it also determines the nilpotency class and the minimal number of generators of the Lie algebra.