Representation of Lie algebras

In mathematics, if φ: G→H is a homomorphism of Lie groups, and g and h are the Lie algebras of G and H respectively, then the induced map φ_* on tangent spaces is a homomorphism of Lie algebras, i.e. satisfies

for all x and y in g. In particular, a representation of Lie groups φ: G→GL(V) determines a homomorphism of Lie algebras from g to the Lie algebra of GL(V), which is just the endomorphism ring End(V) = Hom(V,V). Such a homomorphism is called a representation of the Lie algebra g.

Equivalently, such a representation may be described as a bilinear map (x,v)→x.v from g×V to V satisfying the Jacobi identity analogue

Equivalently, it's a representation of the universal enveloping algebra.

See also representations of Lie groups.