Subrepresentation of a Lie algebra

Definition

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

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

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. When $\mathfrak g$ is semisimple, Weyl’s complete reducibility theorem (see Weyl’s theorem on complete reducibility) says every subrepresentation has an invariant complement, so finite-dimensional representations split as direct sums rather than forming nontrivial extensions.