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Lie Groups, Physics, and Geometry by Robert Gilmore

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8

Structure theory for Lie algebras

 

 

In this chapter we discuss the structure of Lie algebras. A typical Lie algebra is a semidirect sum of a semisimple Lie algebra and a solvable subalgebra that is invariant. By inspection of the regular representation “in suitable form,” we are able to determine the maximal nilpotent and solvable invariant subalgebras of the Lie algebra and its semisimple part. We show how to use the Cartan–Killing inner product to determine which subalgebras in the Lie algebra are nilpotent, solvable, semisimple, and compact.

8.1 Regular representation

A Lie algebra is defined by its commutation relations. The commutation relations are completely encapsulated by the structure constants. These are conveniently summarized ...

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