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More Lie algebra constructions
*July 28, 2009*

*Posted by Akhil Mathew in algebra, representation theory.*

Tags: bilinear forms, Lie algebras, Lie groups, representation theory

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Tags: bilinear forms, Lie algebras, Lie groups, representation theory

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The ultimate aim in the series on Lie algebras I am posting here is to cover the representation theory of semisimple Lie algebras. To get there, we first need to discuss some technical tools—for instance, invariant bilinear forms.

** Generalities on representations **

Fix a Lie algebra . Given representations , we clearly have a representation ; given a morphism of representations , i.e. one which respects the action of , the kernel and image are themselves representations.

Proposition 1The category of finite-dimensional representations of is an abelian category.