Highest-weight representation

Let $\mathfrak g$ be a complex semisimple Lie algebra with a fixed Cartan subalgebra $\mathfrak h$ and choice of positive roots, giving a triangular decomposition

as in the root space decomposition setup.

Definition (Highest-weight representation).
A $\mathfrak g$-module $V$ is a highest-weight representation if there exists a nonzero vector $v\in V$ such that

• $\mathfrak n^+\cdot v=0$, and
• $v$ is a weight vector for $\mathfrak h$, say $h\cdot v=\lambda(h)v$,

and $V$ is generated by $v$ as a $\mathfrak g$-module. The weight $\lambda$ is then the highest weight of $V$.

If $V$ is finite-dimensional and irreducible (see irreducibility ), then $V$ is automatically highest-weight for any choice of positive roots, and the highest weight is uniquely determined by $V$.

Motivation.
Highest-weight representations provide a “triangular” way to build modules: one starts from a single extremal vector and generates everything by applying negative root operators. This is the mechanism behind the classification in the highest-weight theorem .