Irreducible Lie module

Summary

An irreducible module over a Lie algebra $\mathfrak{g}$ is a nonzero representation with no submodules other than $0$ and itself. Irreducible modules are the building blocks of representation theory of Lie algebras, in the same way as irreducible representations of groups. They are the simple modules over the universal enveloping algebra. For a complex semisimple Lie algebra, the finite-dimensional irreducible modules are classified by their highest weights.