# Lie group and Lie algebra representations

A Lie group action $${\rho}$$ is a smooth homomorphism from $${G}$$ to $${\textrm{Diff}(M)}$$. An element of $${G}$$ near the identity then moves each point of $${M}$$ to a nearby point. So for any vector $${A\in\mathfrak{g}}$$, the one-parameter subgroup $${\phi_{A}}$$ from the identity along $${A}$$ maps to a curve in $${M}$$ at each point. The differential of this mapping takes $${A}$$ to a vector field on $${M}$$, and this relation is in fact a Lie algebra homomorphism $${\mathrm{d}\rho\colon\mathfrak{g}\to\textrm{vect}(M)}$$, the corresponding Lie algebra action. Recalling the construction of the Lie derivative, we see that the Lie algebra action of $${A\in\mathfrak{g}}$$, called the fundamental vector field corresponding to $${A}$$, is the vector field on $${M}$$ whose local flow is the Lie group action of the one-parameter subgroup $${\phi_{A}}$$. If $${G}$$ acts on itself by right translation, the fundamental vector fields are just the left-invariant vector fields.

In the case of a Lie group representation on a real or complex vector space $${V}$$, the corresponding Lie algebra representation maps $${\mathfrak{g}}$$ to a linear subalgebra of $${\textrm{vect}(V)}$$ that is isomorphic to $${gl(V)}$$. The Lie bracket in this case is the Lie commutator, whether viewed as that of vector fields, of transformations, or of matrices. Similarly one can show that if a Lie algebra $${\mathfrak{g}}$$ has a matrix representation, and a compact connected Lie group $${G}$$ corresponds to $${\mathfrak{g}}$$, then $${G}$$ has a matrix representation given by the matrix exponential of the Lie algebra representation.

Every finite-dimensional real Lie algebra has a faithful finite-dimensional real representation, i.e. can be viewed as a class of real matrices. This result is a special case of two theorems dealing with scalars in more general fields, Ado’s theorem and Iwasawa’s theorem. The analog is not true in general for finite-dimensional Lie groups, although most Lie groups used in physics can be viewed as matrix groups. A standard counter-example given is the universal covering group of $${SL(2,\mathbb{R})}$$, which is infinite-sheeted and therefore has no faithful finite-dimensional representation.