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.
Read more about this topic: Lie Algebra Representation
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