An Exploration of Lie Theory

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Authors
Cain, Ryan
Advisor
Wendt, Theodore
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Date of Issue
2024
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Title
An Exploration of Lie Theory
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Presentation
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Abstract
Lie theory involves the study of Lie groups, Lie algebras, and their representations. A Lie group is a manifold G, equipped with a smooth group operation × such that G × G --> G. Some of these groups, such as the orthogonal On(K), special linear SLn(K), special orthogonal SO(n), and special unitary SU(n) groups where K is in {R, C, H}, can be represented as matrices G with G as a subset of the general linear group GLn(R). Generally, such representations of G on a Euclidean space Rn are defined as a group homomorphism from G to GLn(R). A Lie algebra g of these matrix groups G is defined as the tangent space to G at the identity I, denoted g(G)=TI(G). These concepts bridge algebra and geometry, and show up in many places in physics, particularly when studying systems with continuous symmetry.
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Mathematics