Subrepresentation of a Lie algebra

Let $\mathfrak g$ be a Lie algebra and let $\rho:\mathfrak g\to \mathfrak{gl}(V)$ be a representation of $\mathfrak g$ on a vector space $V$ , possibly infinite-dimensional. A linear subspace $W\subseteq V$ is a subrepresentation (or $\mathfrak g$ -submodule) if it is invariant under the action:

\rho(X)(W)\subseteq W \quad \text{for all } X\in\mathfrak g.

In this case, restricting $\rho(X)$ to $W$ defines a representation $\rho|_W:\mathfrak g\to \mathfrak{gl}(W)$ .

Quotients and irreducibility

If $W\subseteq V$ is a subrepresentation, then the quotient space $V/W$ inherits a natural $\mathfrak g$ -action via

X\cdot (v+W) = (X\cdot v)+W,

well-defined precisely because $W$ is invariant. A representation is irreducible (see irreducible representations) if its only subrepresentations are $\{0\}$ and $V$ .

Why this matters

Subrepresentations are the “building blocks” for decomposing representations. For a finite-dimensional representation of a semisimple Lie algebra over a field of characteristic zero, Weyl’s complete reducibility theorem says every subrepresentation has an invariant complement. This conclusion does not extend to arbitrary infinite-dimensional modules.