Highest-weight modules and Verma modules

The positive roots determine n+ and the Borel subalgebra b=h⊕n+. The opposite root spaces form n−. For a weight λ in h*, define a one-dimensional b-module Cλ on which h acts by λ and n+ acts by zero. The universal enveloping algebra U(g) is the associative algebra encoding the relation xy−yx=[x,y]. Inducing Cλ from b to g produces the Verma module M(λ). Its distinguished generator is killed by every positive-root operator and has Cartan weight λ.

The Poincaré–Birkhoff–Witt theorem gives a vector-space identification of M(λ) with U(n−). Ordered monomials in negative-root vectors therefore form a basis on its highest generator. Since n− is nonzero, arbitrarily long lowering monomials exist, so every such Verma module is infinite dimensional, even when λ is integral and dominant. Each weight lies below λ by a nonnegative combination of positive roots, and individual weight spaces are finite dimensional. These facts depend on the chosen positive system, not on an unstated global maximum in every direction.

Every module generated by a highest vector of weight λ is a quotient of M(λ), by the universal property of induction. A proper submodule cannot contain its one-dimensional top weight space, because that space generates the whole module. Summing proper submodules still misses that top space, leading to a unique maximal proper submodule. The resulting irreducible quotient is denoted L(λ). The quotient can impose lowering relations that are absent from the Verma module, which is precisely how finite-dimensional simple modules arise at dominant integral weights.

For sl(2), write vⱼ=fʲv₀ for every integer j≥0 in M(m). The same commutator calculation gives evⱼ=j(m−j+1)vⱼ₋₁, but there is no automatic endpoint. If m is a nonnegative integer, vₘ₊₁ is another highest vector because its raising coefficient vanishes. It generates a proper submodule, and quotienting by it leaves the finite string of dimension m+1. Before that quotient, calling M(m) the finite spin module would erase infinitely many basis vectors and the nontrivial submodule structure.

A Verma module can be reducible without splitting as a direct sum of its submodule and quotient. Finite-dimensional complete reducibility for semisimple g does not extend to this infinite-dimensional category. Submodule inclusions, singular vectors, and central characters become meaningful new data. They underpin category O, whose objects satisfy additional finite-generation, weight-space, and local raising-action conditions. Introducing that category requires its hypotheses; merely calling a module highest weight does not prove every categorical property one might want.

For an arbitrary complex λ, L(λ) exists but need not be finite dimensional. In rank one, M(m) is irreducible when m is not a nonnegative integer; the coefficient formula provides the key test. In higher rank, different root directions can produce more elaborate singular vectors, so one rank-one test does not replace the full submodule theory. The practical lesson is to record the Borel, the parameter, the universal construction, and the quotient relations separately before computing dimensions or applying a finite-dimensional character formula.