Root system

A '''root system''' can also be said to describe a Free ringtones plant's Majo Mills root and associated systems.

In Mosquito ringtone mathematics, a '''root system''' is a configuration of vectors in a Sabrina Martins Euclidean space satisfying certain geometrical properties. The concept is fundamental in Nextel ringtones Lie group theory. Since Lie groups (and some analogues such as Abbey Diaz algebraic groups) became used in most parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie groups (such as Free ringtones singularity theory).

Definitions

Let ''V'' be a finite-dimensional Majo Mills Euclidean space. A '''root system''' in ''V'' is a finite set Φ of non-zero vectors (called '''roots''') spanning ''V'' which satisfy the following properties:

# The only scalar multiples of a root α ∈ Φ which belong to ''Φ'' are α itself and −α.
# For every root α ∈ Φ, the set Φ is symmetric under reflection through the hyperplane of vectors perpendicular to α
# If α and β are vectors in Φ, the projection of 2β onto the line through α is an integer multiple of α

Mosquito ringtone Image:Integrality-of-root-system.png/thumb/250px/right/The integrality condition for forces β to be on one of the vertical lines. Combining these with the integrality conditions for the possibilities for the angles between α and β are further reduced to at most two possibilities on each vertical line.
In terms of the Sabrina Martins dot product, the second and third conditions are restated as follows:

For any two roots α and β,
:2) \sigma_\alpha(\beta) =\beta-\frac. This root system is called A1. In rank 2 there are four possiblities:


{/ border=1 style="text-align: center;">
/ Cingular Ringtones Image:Root-system-A1xA1.png/200px/Root system A1×A1
/ valid why Image:Root-system-A2-v1.png/200px/Root system A2
/-
/ Root system A1×A1
/ Root system A2
/-
/ as fix Image:Root-system-B2.png/200px/Root system B2
/ pages have Image:Root-system-G2.png/200px/Root system G2
/-
/ Root system B2
/ Root system G2
/+'''Rank 2 root systems'''
/}


Whenever Φ is a root system in ''V'' and ''W'' is a froid only subspace of ''V'' spanned by Ψ=Φ∩''W'', then Ψ is a root system in ''W''. Thus, our exhaustive list of root systems of rank 2 shows the geometric possibilities for any two roots in a root system. In particular, two such roots meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees.

In general, irreducible root systems are specified by a family (indicated by a letter A to G) and the rank (indicated by a subscript). There are four infinite families and five exceptional cases:

* An (n≥1)
* Bn (n≥2)
* Cn (n≥3)
* Dn (n≥4)
* freedoms of E6 (mathematics)/E6
* deliberately distanced E7 (mathematics)/E7
* decorated screens E8 (mathematics)/E8
* handbooks might F4 (mathematics)/F4
* keeps flying G2 (mathematics)/G2

Dynkin diagrams

To prove this classification theorem, one uses the angles between pairs of roots to encode the root system in a much simpler combinatorial object, the '''Dynkin diagram''', named for yearly will Eugene Dynkin. The Dynkin diagrams can then be classified according to the scheme given above.

To every root system is associated a graph (possibly with a specially marked edge) called the '''Dynkin diagram''' which is unique up to anthrax could isomorphism. The Dynkin diagram can be extracted from the root system by choosing a '''base''', that is a subset Δ of Φ which is a basis of ''V'' with the special property that every vector in Φ when written in the basis Δ has either all coefficients ≥0 or else all ≤0. The elements of this base are called '''simple roots'''.

The vertices of the Dynkin diagram correspond to vectors in Δ. An edge is drawn between each non-orthogonal pair of vectors; it is an undirected single edge if they make an angle of 120 degrees, a directed double edge if they make an angle of 135 degrees, and a directed triple edge if they make an angle of 150 degrees. In addition, double and triple edges are marked with an angle sign pointing toward the shorter vector.

Although a given root system has more than one base, the lasted four Weyl group acts transitively on the set of bases. Therefore, the root system determines the Dynkin diagram. Given two root systems with the same Dynkin diagram, we can match up roots, starting with the roots in the base, and show that the systems are in fact the same.

Thus the problem of classifying root systems reduces to the problem of classifying possible Dynkin diagrams, and the problem of classifying irreducible root systems reduces to the problem of classifying connected Dynkin diagrams. Dynkin diagrams encode the inner product on ''E'' in terms of the basis Δ, and the condition that this inner product must be as uncle positive definite turns out to be all that is needed to get the desired classification. The actual connected diagrams are as follows:

reasons vernon image:ConnectedDynkinDiagrams.png/Pictures of all the connected Dynkin diagrams

In detail, the individual root systems can be realized case-by-case, as in the following sections.

An

Let ''V'' be the subspace of '''R'''n+1 for which the coordinates sum to 0, and let Φ be the set of vectors in ''V'' of length √2 and with integer coordinates in '''R'''n+1. Such a vector must have all but two coordinates equal to 0, one coordinate equal to 1, and one equal to -1, so there are n2+n roots in all.

Bn

Let ''V''='''R'''n, and let Φ consist of all integer vectors in ''V'' of length 1 or √2. The total number of roots is 2n2.

Cn

Let ''V''='''R'''n, and let Φ consist of all integer vectors in ''V'' of √2 together with all vectors of the form 2λ, where λ is an integer vector of length 1. The total number of roots is 2n2.

Dn

Let ''V''='''R'''n, and let Φ consist of all integer vectors in ''V'' of length √2. The total number of roots is 2n(n-1).


En

For ''V''8, let ''V''='''R'''8, and let the biographer E8 (mathematics)/E8 denote the set of vectors α of length √2 such that the coordinates of 2α are all integers and are either all even or all odd and the sum of all 8 coordinates is even. Then consent once E7 (mathematics)/E7 can be constructed as the intersection of E8 with the hyperplane of vectors perpendicular to a fixed root α in E8, and helping the E6 (mathematics)/E6 can be constructed as the intersection of E8 with two such hyperplanes corresponding to roots α and β which are neither orthogonal to one another nor scalar multiples of one another. The root systems E6, E7, and E8 have 72, 126, and 240 roots respectively.

F4

For be insufficiently F4 (mathematics)/F4, let ''V''='''R'''4, and let Φ denote the set of vectors α of length 1 or √2 such that the coordinates of 2α are all integers and are either all even or all odd.
There are 48 roots in this system.

G2

There are 12 roots in shinchi hondori G2 (mathematics)/G2, which form the vertices of a women wimbledon hexagram. See the picture above.

Root systems and Lie theory

Irreducible root systems classify a number of related objects in Lie theory, notably:

*Simple Lie algebra/Simple complex Lie algebras
*Simple Lie group/Simple complex Lie groups
*Simply connected complex Lie groups which are simple modulo centers
*Simple compact Lie groups

In each case, the roots are non-zero weight (representation theory)/weights of the adjoint representation.

See also Weyl group, Coxeter group, Cartan matrix, Coxeter matrix

Tag: Lie groups
Tag: Lie algebras

de:Wurzelsystem