Lie Algebra Representation - Infinitesimal Lie Group Representations

Infinitesimal Lie Group Representations

If φ: G → H is a homomorphism of Lie groups, and and are the Lie algebras of G and H respectively, then the induced map on tangent spaces is a Lie algebra homomorphism. In particular, a representation of Lie groups

determines a Lie algebra homomorphism

from to the Lie algebra of the general linear group GL(V), i.e. the endomorphism algebra of V.

A partial converse to this statement says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated simply connected Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras.