Weight space

Summary

A weight space of a representation of a Lie algebra is a common eigenspace of a family of commuting operators. For a linear form $\lambda$ on a subalgebra $\mathfrak{h}$, it is the set of vectors $v$ with $h \cdot v = \lambda(h) v$ for all $h \in \mathfrak{h}$. A finite-dimensional representation of a complex semisimple Lie algebra is the direct sum of its weight spaces for a Cartan subalgebra. The roots of a Lie algebra are the nonzero weights of its adjoint representation.