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Lie Groups and Lie Algebras

. A Lie group is a smooth manifold that also serves as a group, where the group operations (multiplication and inversion) are smooth maps. The associated Lie algebra is a vector space equipped with a bilinear operation called the Lie bracket, which captures the local structure of the group near the identity element. This connection allows us to study the global properties of Lie groups through their local behavior described by Lie algebras. Key concepts include exponential maps, which link Lie algebras to Lie groups, and the classification of simple Lie algebras, which are fundamental building blocks for understanding more complex structures. These ideas are pivotal in areas like differential geometry, theoretical physics (especially in the study of symmetries), and representation theory.

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