Editing Root system (section)

===Root systems arising from semisimple Lie algebras===
{{See also|Semisimple Lie algebra#Cartan subalgebras and root systems|Root system of a semi-simple Lie algebra}}
If <math>\mathfrak{g}</math> is a complex [[semisimple Lie algebra]] and <math>\mathfrak{h}</math> is a [[Cartan subalgebra]], we can construct a root system as follows. We say that <math>\alpha\in\mathfrak{h}^*</math> is a '''root''' of <math>\mathfrak{g}</math> relative to <math>\mathfrak{h}</math> if <math>\alpha\neq 0</math> and there exists some <math>X\neq 0\in\mathfrak{g}</math> such that
<math display=block>[H,X]=\alpha(H)X</math>
for all <math>H\in\mathfrak{h}</math>. One can show<ref>{{harvnb|Hall|2015|loc=Section 7.5}}</ref> that there is an inner product for which the set of roots forms a root system. The root system of <math>\mathfrak{g}</math> is a fundamental tool for analyzing the structure of <math>\mathfrak{g}</math> and classifying its representations. (See the section below on Root systems and Lie theory.)