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.
","abstract_html":"<p>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.</p> <p>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.</p> <p>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.</p>","abstract_has_math":false,"creators":["Frazier, Avrila"],"institution":null,"degree_name":"Master of Science - Mathematical Sciences","degree_level":"Thesis","degree_discipline":"Mathematics and Statistics","degree_department":null,"school":null,"contributors":["Thomas Judson","Jane Long","Sarah Stovall"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-12-01T08:00:00Z","date_published":"2023-12-01T08:00:00Z","updated_at":"2026-07-24T04:30:45Z","subjects":["Lie algebras","representations","history","simple","classical","exceptional","Algebra"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.sfasu.edu/etds/520","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Thomas Judson","Jane Long","Sarah Stovall"]},{"key":"dc:creator","label":"Author","values":["Frazier, Avrila"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2023-12-05T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics and Statistics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science - Mathematical Sciences"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Lie algebras","representations","history","simple","classical","exceptional","Algebra"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.sfasu.edu/etds/520"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
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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.
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