System of roots

In Mathematical, a system of roots is a configuration of Vecteur S in a Euclidean Espace which checks certain geometrical conditions. This concept is very important in the theory of groups of Dregs. As the groups of Dregs and the algebraic groups are now used in the majority of the parts of mathematics during the twentieth century, the apparently special nature of the systems of roots is in contradiction with the numbers of places in which they are applied. In addition, the diagram of classification of the systems of roots, by the diagrams of Dynkin, appears in parts of mathematics without any manifest connection with the groups of Dregs (such as the Théorie of the singularities).

Definitions

Either V a Euclidean Espace of finished size, provided with the Produces scalar Euclidean standard noted (·, ·). A system of roots in V is a unit finished $\ Phi \,$ of nonnull vectors (called roots) which satisfy the following properties:

the roots generate V like vector space.
only the multiple scalars of a root $\ alpha \ in \ Phi \,$ which are in $\ Phi \,$ are $\ alpha \,$ itself and its opposite $- \ alpha \,$ .
For each root $\ alpha \ in \ Phi \,$ the unit $\ Phi \,$ is stable by the reflection through the Hyperplan perpendicular to $\ alpha \,$ i.e for all roots $\ alpha \,$ and $\ beta \,$ one has,
: $\ sigma_ \ alpha (\ beta) = \ beta-2 \ frac {(\ alpha, \ beta)}{(\ alpha, \ alpha)}\ alpha \ in \ Phi.$
( condition of integrality ) If $\ alpha \,$ and $\ beta \,$ is roots in $\ Phi \,$ , then the orthogonal Projection of $\ beta \,$ on the line generated by $\ alpha \,$ is an half-integral multiple of $\ frac {\ alpha} {2} \,$ :
: $\ langle \ beta, \ alpha \ rangle = 2 \ frac {(\ alpha, \ beta)}{(\ alpha, \ alpha)} \ in \ mathbb {Z},$

Because of property 3, the condition of integrality is equivalent to the following statement: $\ beta \,$ and its image $\ sigma_ {\ alpha} (\ beta) \,$ by the reflection compared to $\ alpha \,$ differs by a multiple entirety from $\ alpha \,$ .

The row of a system of roots $\ Phi \,$ is the Dimension of V . One can combine two systems of roots by making the direct Somme subjacent Euclidean spaces and by taking the union of the roots. A system of roots which cannot be obtained in this manner is known as irreducible .

Two systems of roots $(E_1, \ Phi_1) \,$ and $(E_2, \ Phi_2) \,$ are regarded as identical if there exists a bijection between $E_1 \ rightarrow E_2 \,$ who sends $\ Phi_1 \,$ on $\ Phi_2 \,$ and preserves the reports/ratios of distances.

The group of the Isométrie S of V generated by the reflections compared to the hyperplanes associated with the roots of $\ Phi \,$ is named the Groupe of Weyl of $\ Phi \,$ . As it acts accurately on the unit finished $\ Phi \,$ , the group of Weyl is always finished.

Classification

There exists only one system of roots of row 1 made up of two vectors different from zero ${\ alpha, - \ alpha} \,$ . This system of roots is called $A_1 \,$ . In row 2, there exist four possibilities:

\ Phi \,

is a system of roots in V and W is a Sous-espace of V crossed by

\ Psi= \ Phi \ course W \,

, then

\ Psi \,

is a system of roots in W . Thus, our exhaustive list of system of roots of row 2 shows the geometrical possibilities for two unspecified roots in the system of roots. In particular, two roots of this kind meet with an angle of 0,30,45,60,90,120,135,150 or 180 degrees.

In general, the irreducible systems of roots are specified by a family (indicated by a letter of has to G) and the row (indicated by an index). There exist four family infinite (called the traditional systems of roots ) and five exceptional cases (the exceptional systems of roots ):

$A_n (N \ Ge 1) \,$
$B_n (N \ Ge 2) \,$
$C_n (N \ Ge 3) \,$
$D_n (N \ Ge 4) \,$
E₆
E₇
E₈
F₄
G₂

Positive roots and simple roots

Being given a system of roots $\ Phi \,$ , we can always choose (of many manners) a whole of positive roots . It is a subset $\ Phi^+ \,$ of $\ Phi \,$ such as

for each root $\ alpha \ in \ Phi \,$ exactly one of the roots $\ alpha, - \ alpha \,$ is contained in $\ Phi^+ \,$
For all $\ alpha, \ beta \ in \ Phi^+ \,$ such as $\ alpha+ \ beta \,$ is a root, $\ alpha+ \ beta \ in \ Phi^+ \,$ .

If a whole of positive roots $\ Phi^+ \,$ is chosen, the elements of ( $- \ Phi^+ \,$ ) are called negative roots .

The choice of $\ Phi^+ \,$ is equivalent to the choice of the simple roots . The whole of the simple roots is a subset $\ Delta \,$ of $\ Phi \,$ which is a base of V with the special property that each vector in $\ Phi \,$ when he is written in the base $\ Delta \,$ has is all the coefficients ≥ 0 or all ≤ 0.

It can be shown that for each choice of positive roots, there exists a single whole of simple roots, i.e. the positive roots are exactly these roots which can be expressed like a combination of simple roots with not-negative coefficients.

Stamp of Cartan

Being given the system of simple roots

(\ alpha_i) _ {i=1 \ ldots R}

(where

r

is the row of the system of roots) one defines the matrix of Cartan

A= (a_ {ij}) _ {1 \ Leq I, J \ Leq R}

$a_ {ij} \ equiv \ langle \ alpha_i, \ alpha_j \ rangle = 2 \ frac {(\ alpha_i, \ alpha_j)}{(\ alpha_i, \ alpha_i)}$

The advantage of the matrix of Cartan is that its only data is sufficient to rebuild the whole of all the system of roots. It is thus a theoretical and practical way very useful to code the whole of the contained information in system of roots. To represent the matrix of Cartan graphically one uses the concept of diagram of Dynkin which one now will approach.

Diagrams of Dynkin

To show this theorem of classification, one can use the angles between the pairs of roots for encoder the system of roots in a simpler combinative object, the diagram of Dynkin , named in the honor of Eugene Dynkin. The diagrams of Dynkin can then be classified according to arrangement given above.

With each system of roots is associated a graph (probably with an edge particularly marked) called the diagram with Dynkin which is single except for a Isomorphisme. The diagram of Dynkin can be extracted the system of roots by choosing a unit of simple roots.

The tops of the diagram of Dynkin correspond to the vectors in $\ Delta \,$ . An edge is drawn between each pair of nonorthogonal vectors; there is only one not directed edge if they form an angle of 120 degrees, a double directed edge if they form an angle of 135 degrees and triple directed edge if they form an angle of 150 degrees. Moreover, the doubles and triple edges are marked with a sign of angle pointing towards the shortest vector.

Although a system of roots given has more than one base, the Groupe of Weyl acts transitively on the whole of the bases. Consequently, the system of roots determines the diagram of Dynkin. Being given two systems of roots with the same diagram of Dynkin, we can make coincide the roots, starting with the roots in the base, and go up that the systems are in fact the same ones.

Thus, the problem of classification of the systems of roots is reduced to the problem classification possible diagrams of Dynkin, and the problem of classification of the irreducible systems of roots is reduced to the problem classification connected diagrams of Dynkin. The diagrams of Dynkin encodent the interior product on E in basic terms $\ Delta \,$ , and the condition that this internal product must be positive Défini proves to be all that is necessary to obtain desired classification. The real connected diagrams are the following:

List irreducible systems of roots

The following table lists certain properties of the irreducible systems of roots. Explicit constructions of these systems are given in the following parts.

Here $|\ Phi^ {<}|\,$ indicates the number of short roots (if all the roots have the same length, they are taken as long by definition), I indicates the determinant of the Matrice of Cartan, and $|W|$ indicates the order of the Groupe of Weyl, i.e the number of symmetries of the system of roots.

A_N

Either V , the subspace of $\ mathbb {R} ^ {n+1} \,$ for which the sum of the coordinates equalizes 0, and or $\ Phi \,$ , the whole of the vectors in V length $\ sqrt 2 \,$ and which are whole vectors , i.e which have whole coordinates in $\ mathbb {R} ^ {n+1} \,$ . Such a vector must have all its coordinates except two equal to 0, a coordinate equalizes to 1 and one equalizes with - 1, therefore, there exists $n^2+n$ roots in all.

B_N

Either $V= \ mathbb {R} ^n$ and or $\ Phi \,$ made up of all the whole vectors in V length 1 or $\ sqrt 2 \,$ . The full number of roots is $2n^2$ .

C_N

Either $V= \ mathbb {R} ^n$ and or $\ Phi \,$ made up of all the whole vectors in V length $\ sqrt 2 \,$ at the same time as all the vectors of the form $2 \ lambda \,$ , where $\ lambda \,$ is a whole vector length 1. The full number of roots is $2n^2$ .

D_N

Either $V= \ mathbb {R} ^n$ and or $\ Phi \,$ made up of all the whole vectors in V length $\ sqrt 2 \,$ . The full number of roots is $2n (n-1)$ .

E₆, E₇, E₈

That is to say $V= \ mathbb {R} ^8$ . E_{'' 8 ''} indicates the whole of the vectors $\ alpha \,$ length $\ sqrt 2 \,$ such as the coordinates of $2 \ alpha \,$ are all whole, all pairs or all odd, and such that the sum of the 8 coordinates is even.

With regard to E₇, it can be built like the intersection of E₈ with the hyperplane of vectors perpendicular to a fixed root $\ alpha \,$ in E₈.

Finally, E₆ can be built like the intersection of E₈ with two such hyperplanes, corresponding to the roots $\ alpha \,$ and $\ beta \,$ which are neither orthogonal with another, nor of the multiple scalars to another.

The systems of roots E₆, E₇ and E₈ have respectivemet 72,126 and 240 roots.

F₄

For F₄, either $V= \ mathbb {R} ^4$ , and or $\ Phi \,$ indicating the whole of vectors $\ alpha \,$ length 1 or $\ sqrt 2 \,$ such as the coordinates of $2 \ alpha \,$ is all whole and is or all even or all odd. There exist 48 roots in this system.

G₂

There exist 12 roots in G₂, which form the tops of a Hexagramme. See the image above.

Systems of roots and theory of Dregs

The systems of roots classify a number of objects connected in the theory of Dregs, in particular:

the simple algebras of Dregs complex
the simple groups of Dregs complex
the complex groups of related Dregs Simplement S which are simple modulos their center
the compact groups of Dregs simple

In each case, the roots are the weights different from zero of the assistant Représentation.

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