Main Page | See live article | Alphabetical index 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. This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.