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
Famous quotes containing the words lie and/or group:
“[T]he minister preached a sermon on Jonah and the whale, at the end of which an old chief arose and declared, “We have heard several of the white people talk and lie; we know they will lie, but this is the biggest lie we ever heard.””
—Administration in the State of Miss, U.S. public relief program (1935-1943)
“Unless a group of workers know their work is under surveillance, that they are being rated as fairly as human beings, with the fallibility that goes with human judgment, can rate them, and that at least an attempt is made to measure their worth to an organization in relative terms, they are likely to sink back on length of service as the sole reason for retention and promotion.”
—Mary Barnett Gilson (1877–?)