Knowlpedia

Root space decomposition

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

Let g\mathfrak g be a finite-dimensional complex and let h⊂g\mathfrak h\subset\mathfrak g be a . For each α∈h∗\alpha\in\mathfrak h^*, define

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

and let Φ⊂h∗\Phi\subset\mathfrak h^* be the set of nonzero α\alpha with gα≠0\mathfrak g_\alpha\neq0; these are the .

The root space decomposition is the direct sum decomposition

g  =  h  ⊕  ⨁α∈Φgα.\mathfrak g \;=\; \mathfrak h \;\oplus\; \bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha.

Conceptually, it is the simultaneous eigenspace decomposition for the commuting family {ad⁡(H)}H∈h\{\operatorname{ad}(H)\}_{H\in\mathfrak h} from the .

Two structural bracket relations are fundamental:

  • [h,gα]⊆gα[\mathfrak h,\mathfrak g_\alpha]\subseteq\mathfrak g_\alpha, with [H,X]=α(H)X[H,X]=\alpha(H)X;
  • [gα,gβ]⊆gα+β[\mathfrak g_\alpha,\mathfrak g_\beta]\subseteq\mathfrak g_{\alpha+\beta}, where gγ=0\mathfrak g_\gamma=0 if γ\gamma is not a weight, as explained in .

With the induced by the , Φ\Phi is a . Choosing a yields a triangular decomposition and leads to .