#### Date of Award

12-2017

#### Document Type

Thesis

#### Degree Name

Master of Arts in Mathematics

#### Department

Mathematics

#### First Reader/Committee Chair

Llosent, Giovanna

#### Abstract

An (associative) algebra is a vector space over a field equipped with an associative, bilinear multiplication. By use of a new bilinear operation, any associative algebra morphs into a nonassociative abstract Lie algebra, where the new product in terms of the given associative product, is the commutator. The crux of this paper is to investigate the commutator as it pertains to the general linear group and its subalgebras. This forces us to examine properties of ring theory under the lens of linear algebra, as we determine subalgebras, ideals, and solvability as decomposed into an extension of abelian ideals, and nilpotency, as decomposed into the lower central series and eventual zero subspace. The map sending the Lie algebra L to a derivation of L is called the adjoint representation, where a given Lie algebra is nilpotent if and only if the adjoint is nilpotent. Our goal is to prove Engel's Theorem, which states that if all elements of L are ad-nilpotent, then L is nilpotent.

#### Recommended Citation

Talley, Amanda Renee, "An Introduction to Lie Algebra" (2017). *Electronic Theses, Projects, and Dissertations*. 591.

https://scholarworks.lib.csusb.edu/etd/591