Editing Root system (section)

==Root systems and Lie theory==
Irreducible root systems classify a number of related objects in Lie theory, notably the following:
*[[Simple Lie algebra|simple complex Lie algebras]] (see the discussion above on root systems arising from semisimple Lie algebras),
*[[simply connected]] complex Lie groups which are simple modulo centers, and
*[[simply connected]] [[compact group|compact Lie groups]] which are simple modulo centers.
In each case, the roots are non-zero [[weight (representation theory)|weight]]s of the [[Adjoint representation of a Lie algebra|adjoint representation]].

We now give a brief indication of how irreducible root systems classify simple Lie algebras over <math>\mathbb C</math>, following the arguments in Humphreys.<ref>See various parts of Chapters III, IV, and V of {{harvnb|Humphreys|1972}}, culminating in Section 19 in Chapter V</ref> A preliminary result says that a [[semisimple Lie algebra]] is simple if and only if the associated root system is irreducible.<ref>{{harvnb|Hall|2015}}, Theorem 7.35</ref> We thus restrict attention to irreducible root systems and simple Lie algebras.

*First, we must establish that for each simple algebra <math>\mathfrak g</math> there is only one root system. This assertion follows from the result that the Cartan subalgebra of <math>\mathfrak g</math> is unique up to automorphism,<ref>{{harvnb|Humphreys|1972|loc=Section 16}}</ref> from which it follows that any two Cartan subalgebras give isomorphic root systems. 
*Next, we need to show that for each irreducible root system, there can be at most one Lie algebra, that is, that the root system determines the Lie algebra up to isomorphism.<ref>{{harvnb|Humphreys|1972|loc=Part (b) of Theorem 18.4}}</ref> 
*Finally, we must show that for each irreducible root system, there is an associated simple Lie algebra. This claim is obvious for the root systems of type A, B, C, and D, for which the associated Lie algebras are the [[classical Lie algebras]]. It is then possible to analyze the exceptional algebras in a case-by-case fashion. Alternatively, one can develop a systematic procedure for building a Lie algebra from a root system, using [[Root system of a semi-simple Lie algebra#Serre.27s relations: Associating a semisimple Lie algebra to a root system|Serre's relations]].<ref>{{harvnb|Humphreys|1972}} Section 18.3 and Theorem 18.4</ref>

For connections between the exceptional root systems and their Lie groups and Lie algebras see [[E8 (mathematics)|E<sub>8</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], [[E6 (mathematics)|E<sub>6</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], and [[G2 (mathematics)|G<sub>2</sub>]].