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Lie Groups, Physics, and Geometry by Robert Gilmore

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4

Lie algebras

 

 

The study of Lie groups can be greatly facilitated by linearizing the group in the neighborhood of its identity. This results in a structure called a Lie algebra. The Lie algebra retains most, but not quite all, of the properties of the original Lie group. Moreover, most of the Lie group properties can be recovered by the inverse of the linearization operation, carried out by the EXPonential mapping. Since the Lie algebra is a linear vector space, it can be studied using all the standard tools available for linear vector spaces. In particular, we can define convenient inner products and make standard choices of basis vectors. The properties of a Lie algebra in the neighborhood of the origin are identified with the properties ...

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