Titlebook: Lie Groups, Lie Algebras, and Representations; An Elementary Introd Brian C. Hall Textbook 2015Latest edition Springer International Publis

河潭

https://doi.org/10.1007/978-3-319-13467-3Baker-Campbell-Hausdorff formula; Cartan-Weyl theory; Lie algebras; Lie groups; representation theory

辞职

使乳化

The Matrix ExponentialThe exponential of a matrix plays a crucial role in the theory of Lie groups. The exponential enters into the definition of the Lie algebra of a matrix Lie group (Sect. 3.3) and is the mechanism for passing information from the Lie algebra to the Lie group.

呼吸

寄生虫

Compact Lie Groups and Maximal ToriIn this chapter and Chapter . we develop the representation theory of a connected, compact matrix Lie group .. The main result is a “theorem of the highest weight,” which is very similar to our main results for semisimple Lie algebras.

mydriatic

易受骗

Fundamental Groups of Compact Lie GroupsIn this section, we briefly review the notion of the fundamental group of a topological space. For a more detailed treatment, the reader should consult any standard book on algebraic topology, such as [Hat, Chapter 1]. Let . be any path-connected Hausdorff topological space and let . be a fixed point in . (the “basepoint”).

联想记忆

Cervical-Spine

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Pcos971

Representations of Semisimple Lie Algebraspresentations whose highest weights are the fundamental weights in Definition 8.36. Thus, we require a new method of constructing the irreducible representation of . with a given dominant integral highest weight. This construction will be the main topic of the present chapter.