Representation of a Lie Algebra

A Lie algebra homomorphism from a Lie algebra to endomorphisms of a vector space.

Representation of a Lie Algebra

Let $\mathfrak{g}$ be a Lie algebra and let $V$ be a vector space . A representation of $\mathfrak{g}$ is a Lie algebra homomorphism

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

where $\mathfrak{gl}(V)$ is the Lie algebra of all linear operators on $V$ with bracket $[A,B]=AB-BA$ .

Explicitly, $\rho$ must satisfy

\rho([X,Y]) = [\rho(X),\rho(Y)] \quad \text{for all } X,Y\in\mathfrak{g}.

Equivalent “module” viewpoint

Giving $\rho$ is the same as giving an action $\mathfrak{g}\times V\to V$ , $(X,v)\mapsto X\cdot v$ , such that

[X,Y]\cdot v = X\cdot (Y\cdot v) - Y\cdot (X\cdot v).

The trivial representation: $\rho(X)=0$ for all $X$ .
The adjoint representation $\operatorname{ad}:\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})$ .
Any Lie group representation $\rho:G\to \operatorname{GL}(V)$ differentiates to one of $\mathfrak{g}=T_eG$ .

The kernel of a representation is always an ideal of $\mathfrak{g}$ .