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

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 $\Phi\subset \mathfrak h^*$.

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 .

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 ).