Beyond finite-dimensional highest-weight theory

This page stops at the finite-dimensional theorem of the highest weight and its immediate consequences. The wider representation theory of complex semisimple Lie algebras is deliberately not developed here and is not used anywhere in the finite-dimensional classification above: Verma modules and their simple quotients, the Bernstein–Gelfand–Gelfand category $\mathcal{O}$, the Harish–Chandra isomorphism and the centre of the enveloping algebra, Kazhdan–Lusztig theory, and geometric representation theory all require machinery beyond the scope of this page.

In particular no item above depends on a Verma-module construction, on a character formula, or on any categorical or geometric representation theory. The cyclic quotient $M_{\mathrm{i}\mathrm{n}\mathrm{t}}(\lambda )$ and its finite-dimensional simple quotient $L(\lambda )$ are built directly from the enveloping algebra and the root-space structure of $\mathfrak{g}$, and complete reducibility of finite-dimensional representations is imported from the general theory of semisimple Lie algebras rather than from category $\mathcal{O}$.