Weight of a representation

Summary

A weight of a representation of a Lie algebra is a joint eigenvalue of an abelian subalgebra, recorded as a linear form. For a complex semisimple Lie algebra, one uses a Cartan subalgebra, and every finite-dimensional module is the direct sum of its weight spaces. The weights of the adjoint representation are $0$ and the roots. Weights and their multiplicities organise the whole finite-dimensional representation theory.