Editing Root system (section)

==Dual root system, coroots, and integral elements==
{{See also|Langlands dual group}}

===The dual root system===
If Φ is a root system in ''E'', the '''coroot''' α<sup>∨</sup> of a root α is defined by
<math display=block>\alpha^\vee= {2\over (\alpha,\alpha)}\, \alpha.</math>

The set of coroots also forms a root system Φ<sup>∨</sup> in ''E'', called the '''dual root system''' (or sometimes ''inverse root system'').
By definition,  α<sup>∨ ∨</sup> = α, so that Φ is the dual root system of Φ<sup>∨</sup>. The lattice in ''E'' spanned by  Φ<sup>∨</sup> is called the ''coroot lattice''. Both Φ and Φ<sup>∨</sup> have the same Weyl group ''W'' and, for ''s'' in ''W'',
<math display=block> (s\alpha)^\vee= s(\alpha^\vee).</math>

If Δ is a set of simple roots for Φ, then Δ<sup>∨</sup> is a set of simple roots for Φ<sup>∨</sup>.<ref>{{harvnb|Hall|2015|loc=Proposition 8.18}}</ref>

In the classification described below, the root systems of type <math>A_n</math> and <math>D_n</math> along with the exceptional root systems <math>E_6,E_7,E_8,F_4,G_2</math> are all self-dual, meaning that the dual root system is isomorphic to the original root system. By contrast, the <math>B_n</math> and <math>C_n</math> root systems are dual to one another, but not isomorphic (except when <math>n=2</math>).

===Integral elements===
{{See also|Weight (representation theory)#Weights in the representation theory of semisimple Lie algebras}}
A vector <math>\lambda</math> in ''E'' is called '''integral'''<ref>{{harvnb|Hall|2015|loc=Section 8.7}}</ref> if its inner product with each coroot is an integer:
<math display=block>2\frac{(\lambda,\alpha)}{(\alpha,\alpha)}\in\mathbb Z,\quad\alpha\in\Phi.</math>
Since the set of <math>\alpha^\vee </math> with <math>\alpha\in\Delta</math> forms a base for the dual root system, to verify that <math>\lambda</math> is integral, it suffices to check the above condition for <math>\alpha\in\Delta</math>.

The set of integral elements is called the '''weight lattice''' associated to the given root system. This term comes from the [[Lie algebra representation#Classifying finite-dimensional representations of Lie algebras|representation theory of semisimple Lie algebras]], where the integral elements form the possible weights of finite-dimensional representations.

The definition of a root system guarantees that the roots themselves are integral elements. Thus, every integer linear combination of roots is also integral. In most cases, however, there will be integral elements that are not integer combinations of roots. That is to say, in general the weight lattice does not coincide with the root lattice.