Editing Root system (section)

===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''.