Root space decomposition

Decomposition of a semisimple Lie algebra into a Cartan subalgebra plus root spaces for the adjoint action.

Root space decomposition

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 each $\alpha\in\mathfrak h^*$ define the weight space

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

and let $\Phi\subset\mathfrak h^*$ be the set of nonzero $\alpha$ with $\mathfrak g_\alpha\neq 0$ (the roots ).

The root space decomposition (sometimes called the Cartan decomposition of $\mathfrak g$ ) is the direct sum decomposition

\mathfrak g \;=\; \mathfrak h \;\oplus\; \bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha.

Conceptually, it is the simultaneous eigenspace decomposition for the commuting family of endomorphisms $\{\mathrm{ad}(H)\}_{H\in\mathfrak h}$ coming from the adjoint representation .

Two structural bracket relations are fundamental:

$[\mathfrak h,\mathfrak g_\alpha]\subseteq \mathfrak g_\alpha$ with the eigenvalue rule $[H,X]=\alpha(H)X$ ;
$[\mathfrak g_\alpha,\mathfrak g_\beta]\subseteq \mathfrak g_{\alpha+\beta}$ (with the convention $\mathfrak g_{\gamma}=0$ if $\gamma$ is not a weight), as explained in root spaces .

With the inner product induced by the Killing form , the set $\Phi$ satisfies the axioms of a root system . Choosing a positive system refines this into a triangular decomposition and is the starting point for Dynkin diagram combinatorics (see Dynkin diagrams ).