Irreducible representation of a Lie algebra

Let $\mathfrak g$ be a Lie algebra and let $\rho:\mathfrak g\to \mathfrak{gl}(V)$ be a representation on a nonzero vector space $V$ , possibly infinite-dimensional.

Definition (Irreducible). The representation $(\rho,V)$ is irreducible if the only $\mathfrak g$ -invariant subspaces of $V$ are $\{0\}$ and $V$ . Equivalently, $V$ is a simple $\mathfrak g$ -module.

A subspace $W\subseteq V$ is $\mathfrak g$ -invariant precisely when $\rho(x)W\subseteq W$ for all $x\in\mathfrak g$ ; such a $W$ is a subrepresentation.

Remarks

This is algebraic irreducibility: no topology or closedness condition is imposed on invariant subspaces. An irreducible unitary representation can have an algebraically reducible derived action on its smooth vectors, as the Heisenberg example demonstrates.

Context. Irreducibles are the building blocks for representation theory. For semisimple $\mathfrak g$ , every finite-dimensional representation is completely reducible (see Weyl's theorem and complete reducibility), and irreducibles are classified by the highest-weight theorem.