Representation of a Lie Group

Let $G$ be a Lie group and $V$ a finite-dimensional vector space. A (linear) representation of $G$ is a Lie group homomorphism

\rho: G \to \operatorname{GL}(V),

where $\operatorname{GL}(V)$ is the group of invertible linear operators on $V$ .

Differentiating a representation

The differential at the identity gives a Lie algebra representation

d\rho_e : \mathfrak{g} \to \mathfrak{gl}(V),

which is a representation of the Lie algebra $\mathfrak{g}=T_eG$ ; see Lie algebra of a Lie group and differential.

Examples

The defining representation $\operatorname{GL}(n,\mathbb{R})\to \operatorname{GL}(\mathbb{R}^n)$ .
Orthogonal/unitary representations of $\operatorname{SO}(n)$ , $\operatorname{SU}(n)$ on $\mathbb{R}^n$ , $\mathbb{C}^n$ .
The adjoint representation $G\to \operatorname{Aut}(\mathfrak{g})$ .

Many structural notions for representations can be studied via the induced Lie algebra representation and tools such as the Killing form in the semisimple case.

Equivalent characterizations

Equivalently, $\rho$ is a smooth action of $G$ on $V$ by linear isomorphisms, written $g\cdot v := \rho(g)v$ .