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.