Lie theory MOC

Lie algebra representation

A representation of a Lie algebra is a Lie algebra homomorphism from to the commutator algebra of the Endomorphism ring of . lie Thus for any ,

A space carrying a representation of is sometimes called a -module,¹ though this is an abuse of terminology.

Properties

• Every Lie algebra has a natural representation on itself, the adjoint Lie algebra representation.
• Every representation of a Lie group induces an Infinitesimal representation of the corresponding Lie algebra.
• Sum of commuting Lie algebra representations
• Tensor product of Lie algebra representations

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Footnotes

1. 1988. Vertex operator algebras and the Monster, p. 5 ↩