Root of a Lie algebra

Let $\mathfrak g$ be a finite-dimensional complex semisimple Lie algebra (see semisimple Lie algebra) and let $\mathfrak h\subset \mathfrak g$ be a Cartan subalgebra. For $\alpha\in \mathfrak h^*$ , define the $\alpha$ -weight space for the adjoint action by

\mathfrak g_\alpha \;:=\;\{X\in\mathfrak g : [H,X]=\alpha(H)\,X\ \text{for all }H\in\mathfrak h\}.

A nonzero functional $\alpha\in\mathfrak h^*$ is called a root of $\mathfrak g$ (relative to $\mathfrak h$ ) if $\mathfrak g_\alpha\neq 0$ . The set of roots is denoted $\Phi\subset \mathfrak h^*$ .

Roots are the combinatorial shadow of $\mathfrak g$ : much of the structure and classification of semisimple Lie algebras is encoded in $\Phi$ (compare Dynkin diagrams and classification of simple Lie algebras).

Equivalent characterizations

Equivalently, roots are the nonzero weights of the adjoint representation restricted to $\mathfrak h$ . Each root comes with its root space $\mathfrak g_\alpha$ , and together they assemble into the root space decomposition. With the inner product on $\mathfrak h^*$ induced by the Killing form, the set $\Phi$ satisfies the axioms of a root system.