Date of Award


Degree Type


Degree Name

Master of Science - Mathematical Sciences


Mathematics and Statistics

First Advisor

Thomas Judson

Second Advisor

Jane Long

Third Advisor

Sarah Stovall

Fourth Advisor

Jeremy Becnel


In 1869, prompted by his work in differential equations, Sophus Lie wondered about categorizing what he called “closed systems of commutative transformations,” while around the same time, Wilhelm Killing’s work on non-Euclidean geometry encountered related topics. As mathematicians recognized this as a division of abstract algebra, the area became known as “continuous transformation groups," but we now refer to them as Lie groups.

Patterns and structures emerged from their work, such as describing Lie groups in connection with their associated Lie algebras, which can be categorized in many important ways. In this paper, we focus on Lie algebras over the complex numbers, and how simplicity and the related notion of semisimplicity, as well as root spaces and their representations, reveal that there are, up to isomorphism, surprisingly few simple complex Lie algebras, a result which Killing examined intuitively.

Élie Cartan’s influence on the development of the theory of Lie algebras, though chronologically slightly later, was key to making the theory of Lie algebras the influential topic it continues to be today. He brought the rigor Lie preferred to bear on ideas and patterns generated by Killing; among other impacts of his approach, in solving the classification problem of simple complex Lie algebras.

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

Included in

Algebra Commons



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