Editing Maximal torus (section)

==Root system==
If ''T'' is a maximal torus in a compact Lie group ''G'', one can define a [[root system]] as follows. The roots are the [[Compact_group#Representation_theory_of_a_connected_compact_Lie_group|weights]] for the adjoint action of ''T'' on the complexified Lie algebra of ''G''. To be more explicit, let <math>\mathfrak t</math> denote the Lie algebra of ''T'', let <math>\mathfrak g</math> denote the Lie algebra of <math>G</math>, and let <math>\mathfrak g_{\mathbb C}:=\mathfrak g\oplus i\mathfrak g</math> denote the complexification of <math>\mathfrak g</math>. Then we say that an element <math>\alpha\in\mathfrak t</math> is a '''root''' for ''G'' relative to ''T'' if <math>\alpha\neq 0</math> and there exists a nonzero <math>X\in\mathfrak g_{\mathbb C}</math> such that
:<math>\mathrm{Ad}_{e^H}(X)=e^{i\langle\alpha,H\rangle}X</math>
for all <math>H\in\mathfrak t</math>. Here <math>\langle\cdot,\cdot\rangle</math> is a fixed inner product on <math>\mathfrak g</math> that is invariant under the adjoint action of connected compact Lie groups.

The root system, as a subset of the Lie algebra <math>\mathfrak t</math> of ''T'', has all the usual properties of a root system, except that the roots may not span <math>\mathfrak t</math>.<ref>{{harvnb|Hall|2015}} Section 11.7</ref> The root system is a key tool in understanding the [[Compact_group#Classification|classification]] and [[Compact_group#Representation_theory_of_a_connected_compact_Lie_group|representation theory]] of ''G''.