Editing Root system (section)

==Definitions and examples==
[[File:Root system A2 with labels.png|class=skin-invert-image|right|thumb|250px|The six vectors of the root system ''A''<sub>2</sub>]]
As a first example, consider the six vectors in 2-dimensional [[Euclidean space]], '''R'''<sup>2</sup>, as shown in the image at the right; call them '''roots'''. These vectors [[Linear span|span]] the whole space. If you consider the line [[perpendicular]] to any root, say ''β'', then the reflection of '''R'''<sup>2</sup> in that line sends any other root, say ''α'', to another root. Moreover, the root to which it is sent equals ''α'' + ''nβ'', where ''n'' is an integer (in this case, ''n'' equals 1). These six vectors satisfy the following definition, and therefore they form a root system; this one is known as ''A''<sub>2</sub>.

===Definition===
Let ''E'' be a finite-dimensional [[Euclidean space|Euclidean]] [[vector space]], with the standard [[Dot product|Euclidean inner product]] denoted by <math>(\cdot,\cdot)</math>. A '''root system''' <math>\Phi</math> in ''E'' is a finite set of non-zero vectors (called '''roots''') that satisfy the following conditions:<ref>Bourbaki, Ch.VI, Section 1</ref><ref>{{harvnb|Humphreys|1972|p=42}}</ref>

# The roots [[linear span|span]] ''E''.
# The only scalar multiples of a root <math>\alpha\in\Phi</math> that belong to <math>\Phi</math> are <math>\alpha</math> itself and <math>-\alpha</math>.
# For every root <math>\alpha\in\Phi</math>, the set <math>\Phi</math> is closed under [[Reflection (mathematics)|reflection]] through the [[hyperplane]] perpendicular to <math>\alpha</math>.
# ('''Integrality''') If <math>\alpha</math> and <math>\beta</math> are roots in <math>\Phi</math>, then the projection of <math>\beta</math> onto the line through <math>\alpha</math> is an ''integer or half-integer'' multiple of <math>\alpha</math>.

Equivalent ways of writing conditions 3 and 4, respectively, are as follows:

#<li value="3"> For any two roots <math>\alpha,\beta \in \Phi </math>, the set <math>\Phi</math> contains the element <math>\sigma_\alpha(\beta):=\beta-2\frac{(\alpha,\beta)}{(\alpha,\alpha)}\alpha.</math></li>
# For any two roots <math>\alpha,\beta\in\Phi</math>, the number <math> \langle \beta, \alpha \rangle := 2 \frac{(\alpha,\beta)}{(\alpha,\alpha)}</math> is an [[integer]].

Some authors only include conditions 1&ndash;3 in the definition of a root system.<ref>{{harvnb|Humphreys|1992|p=6}}</ref>  In this context, a root system that also satisfies the integrality condition is known as a '''crystallographic root system'''.<ref>{{harvnb|Humphreys|1992|p=39}}</ref>  Other authors omit condition 2; then they call root systems satisfying condition 2 '''reduced'''.<ref>{{harvnb|Humphreys|1992|p=41}}</ref>  In this article, all root systems are assumed to be reduced and crystallographic.

In view of property 3, the integrality condition is equivalent to stating that ''β'' and its reflection ''σ''<sub>''α''</sub>(''β'') differ by an integer multiple of&nbsp;''α''. Note that the operator
<math display=block> \langle \cdot, \cdot \rangle \colon \Phi \times \Phi \to \mathbb{Z}</math>
defined by property 4 is not an inner product. It is not necessarily symmetric and is linear only in the first argument. 
{| class="wikitable" align="right" width=300
|+'''Rank-2 root systems'''
|- align=center
| [[Image:Root system A1xA1.svg|class=skin-invert-image|150px|Root system A<sub>1</sub> + A<sub>1</sub>]]
| [[Image:Root system D2.svg|class=skin-invert-image|150px|Root system D<sub>2</sub>]]
|- align=center BGCOLOR="#ddd"
| Root system <math>A_1 \times A_1</math><BR>{{Dynkin|node_n1|2|node_n2}}
| Root system <math>D_2</math><BR>{{Dynkin2|nodes}}
|- align=center
| [[Image:Root system A2.svg|class=skin-invert-image|150px|Root system A<sub>2</sub>]]
| [[Image:Root system G2.svg|class=skin-invert-image|150px|Root system G<sub>2</sub>]]
|- align=center BGCOLOR="#ddd"
| Root system <math>A_2</math><BR>{{Dynkin2|node_n1|3|node_n2}}
| Root system <math>G_2</math><BR>{{Dynkin2|nodeg_n1|6a|node_n2}}
|- align=center
| [[Image:Root system B2.svg|class=skin-invert-image|150px|Root system B<sub>2</sub>]]
| [[Image:Root system C2 (fixed).svg|class=skin-invert-image|150px|Root system C<sub>2</sub>]]
|- align=center BGCOLOR="#ddd"
| Root system <math>B_2</math><BR>{{Dynkin2|nodeg_n1|4a|node_n2}}
| Root system <math>C_2</math><BR>{{Dynkin2|node_n1|4b|nodeg_n2}}
|}
The '''rank''' of a root system Φ is the dimension of ''E''.  
Two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space.  A root system which does not arise from such a combination, such as the systems ''A''<sub>2</sub>, ''B''<sub>2</sub>, and ''G''<sub>2</sub> pictured to the right, is said to be '''irreducible'''.

Two root systems (''E''<sub>1</sub>,&nbsp;Φ<sub>1</sub>) and (''E''<sub>2</sub>,&nbsp;Φ<sub>2</sub>) are called '''isomorphic''' if there is an invertible linear transformation ''E''<sub>1</sub>&nbsp;→&nbsp;''E''<sub>2</sub> which sends&nbsp;Φ<sub>1</sub> to&nbsp;Φ<sub>2</sub> such that for each pair of roots, the number <math> \langle x, y \rangle</math> is preserved.<ref>{{harvnb|Humphreys|1972|p=43}}</ref>

The '''{{visible anchor|root lattice}}''' of a root system Φ is the '''Z'''-submodule of ''E'' generated by&nbsp;Φ. It is a [[lattice (discrete subgroup)|lattice]] in&nbsp;''E''.

===Weyl group===
{{Main|Weyl group}}
[[File:A2_Weyl_group_(revised).png|class=skin-invert-image|thumb|right|The Weyl group of the <math>A_2</math> root system is the symmetry group of an equilateral triangle]]
The [[group (mathematics)|group]] of [[isometry|isometries]] of&nbsp;''E'' generated by reflections through hyperplanes associated to the roots of&nbsp;Φ is called the [[Weyl group]] of&nbsp;Φ. As it [[Faithful action|acts faithfully]] on the finite set&nbsp;Φ, the Weyl group is always finite. The reflection planes are the hyperplanes perpendicular to the roots, indicated for <math>A_2</math> by dashed lines in the figure below. The Weyl group is the symmetry group of an equilateral triangle, which has six elements. In this case, the Weyl group is not the full symmetry group of the root system (e.g., a 60-degree rotation is a symmetry of the root system but not an element of the Weyl group).

===Rank one example===

There is only one root system of rank 1, consisting of two nonzero vectors <math>\{\alpha, -\alpha\}</math>. This root system is called <math>A_1</math>.

===Rank two examples===

In rank 2 there are four possibilities, corresponding to <math>\sigma_\alpha(\beta) = \beta + n\alpha</math>, where <math>n = 0, 1, 2, 3</math>.<ref>{{harvnb|Hall|2015}} Proposition 8.8</ref> The figure at right shows these possibilities, but with some redundancies: <math>A_1\times A_1</math> is isomorphic to <math>D_2</math> and <math>B_2</math> is isomorphic to <math>C_2</math>.

Note that a root system is not determined by the lattice that it generates: <math>A_1 \times A_1</math> and <math>B_2</math> both generate a [[square lattice]] while <math>A_2</math> and <math>G_2</math> both generate a [[hexagonal lattice]].

Whenever Φ is a root system in ''E'', and ''S'' is a [[Linear subspace|subspace]] of ''E'' spanned by Ψ&nbsp;=&nbsp;Φ&nbsp;∩&nbsp;''S'', then&nbsp;Ψ is a root system in&nbsp;''S''.  Thus, the exhaustive list of four root systems of rank&nbsp;2 shows the geometric possibilities for any two roots chosen from a root system of arbitrary rank.  In particular, two such roots must meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees.

===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.)