Root system

This article discusses root systems in mathematics. For root systems of plants, see root.

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in Lie group theory. Since Lie groups (and some analogues such as algebraic groups) have come to be used in many 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 singularity theory).

Definitions

Let V be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by . A root system in V is a finite set Φ of non-zero vectors (called roots) that satisfy the following properties:

The integrality condition for <α, β> forces β to be on one of the vertical lines. Combining these with the integrality conditions for <β, α> the possibilities for the angles between α and β are further reduced to at most two possibilities on each vertical line.

1. The roots span V
2. The only scalar multiples of a root α ∈ Φ that belong to Φ are α itself and −α.
3. For every root α ∈ Φ, the set Φ is closed under reflection through the hyperplane perpendicular to α. That is, for any two roots α and β, the set Φ contains the reflection of β,

   

4. (Integrality condition) If α and β are roots in Φ, then the projection of β onto the line through α is a half-integral multiple of α. That is,

   

In view of property 3, the integrality condition is equivalent to stating that β and its reflection σ_α(β) differ by an integer multiple of α. Note that the operator

defined by property 4 is not an inner product. It is not necessarily symmetric and is linear only in the first argument.

The integrality condition also means that the ratio of the lengths (magnitudes) of any two roots cannot be 2 or greater, since otherwise either the projection of the shorter root onto the longer root will be less than half as long as the longer root, or the shorter root will be exactly half the longer root or its negative.

The cosine of the angle between two roots is constrained to be a half-integral multiple of a square root of an integer:

These values can only be , corresponding to angles of 30°, 45°, 60°, 90°, 120°, 135°, 150°.

The rank of a root system Φ is the dimension of V. 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 A₂, B₂, and G₂ pictured below, is said to be irreducible.

Two irreducible root systems (E₁,Φ₁) and (E₂,Φ₂) are considered to be the same if there is an invertible linear transformation E₁→E₂ which sends Φ₁ to Φ₂.

The group of isometries of V generated by reflections through hyperplanes associated to the roots of Φ is called the Weyl group of Φ. As it acts faithfully on the finite set Φ, the Weyl group is always finite.

The root lattice of a root system Φ is the Z-submodule of V generated by Φ. It is a lattice in V.

Rank 1 and rank 2 examples

There is only one root system of rank 1, consisting of two nonzero vectors {α, −α}. This root system is called A₁.

In rank 2 there are four possibilities, corresponding to σ_α(β) = β + nα, where n = 0, 1, 2, 3.

Rank 2 root systems

Root system A1×A1	Root system A2
Root system B2	Root system G2

Whenever Φ is a root system in V and W is a 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 chosen from a root system of arbitrary rank. In particular, two such roots meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees.

Positive roots and simple roots

Given a root system Φ we can always choose (in many ways) a set of positive roots. This is a subset of Φ such that

• for each root exactly one of the roots is contained in
• For any such that is a root, .

If a set of positive roots is chosen, elements of () are called negative roots.

An element of is called indecomposable or simple if it cannot be written as the sum of two elements of . The set of simple roots is a basis of with the property that every vector in is a linear combination of elements of with all coefficients non-negative, or all coefficients non-positive.

It can be shown that for each choice of positive roots there exists a unique set of simple roots so that the positive roots are exactly those roots that can be expressed as a combination of simple roots with non-negative coefficients.

Dual root system and coroots

See also: Langlands dual group

If Φ is a root system in V, the coroot α^V of a root α is defined by

The set of coroots roots also forms a root system Φ^V in V, called the dual root system (or sometimes inverse root system). By definition, α^{V V} = α, so that Φ is the dual root system of Φ^V. The lattice in V spanned by Φ^V is called the coroot lattice. Both Φ and Φ^V have the same Weyl group W and, for s in W,

If Δ is a set of simple roots for Φ, then Δ^V is a set of simple roots for Φ^V.

Classification of root systems by Dynkin diagrams

Irreducible root systems correspond to certain graphs, the Dynkin diagrams named after Eugene Dynkin. The classification of these graphs is a simple matter of combinatorics, and induces a classification of irreducible root systems.

Given a root system, select a set Δ of simple roots as in the preceding section. The vertices of the associated 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. The term "directed edge" means that double and triple edges are marked with an angle sign pointing toward the shorter vector.

Although a given root system has more than one possible set of simple roots, the Weyl group acts transitively on such choices. Consequently, the Dynkin diagram is independent of the choice of simple roots; it is determined by the root system itself. Conversely, given two root systems with the same Dynkin diagram, one 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. 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 positive definite turns out to be all that is needed to get the desired classification.

The actual connected diagrams are as follows. The subscripts indicate the number of vertices in the diagram (and hence the rank of the corresponding irreducible root system).

Properties of the irreducible root systems

			I	D
An (n≥1)	n(n+1)			n+1	(n+1)!
Bn (n≥2)	2n2	2n	2	2	2n n!
Cn (n≥3)	2n2	2n(n−1)	2	2	2n n!
Dn (n≥4)	2n(n−1)			4	2n−1 n!
E6	72			3	51840
E7	126			2	2903040
E8	240			1	696729600
F4	48	24	4	1	1152
G2	12	6	3	1	12

Irreducible root systems are named according to their corresponding connected Dynkin diagrams. There are four infinite families (A_n, B_n, C_n, and D_n, called the classical root systems) and five exceptional cases (the exceptional root systems). The subscript indicates the rank of the root system.

In an irreducible root system there can be most two values for the length (α, α)^½, corresponding to short and long roots. If all roots have the same length they are taken to be long by definition and the root system is said to be simply laced; this occurs in the cases A, D and E. Any two roots of the same length lie in the same orbit of the Weyl group. In the non-simply laced cases B, C, G and F, the root lattice is spanned by the short roots and the long roots span a sublattice, invariant under the Weyl group, equal to r²/2 times the coroot lattice, where r is the length of a long root.

In the table to the right, denotes the number of short roots, I denotes the index in the root lattice of the sublattice generated by long roots, D denotes the determinant of the Cartan matrix, and denotes the order of the Weyl group.

Explicit construction of the irreducible root systems

A_n

Let V be the subspace of Rⁿ⁺¹ for which the coordinates sum to 0, and let Φ be the set of vectors in V of length √2 and which are integer vectors, i.e. have integer coordinates in Rⁿ⁺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. One choice of simple roots expressed in the standard basis is: α_i = e_i - e_i+1, for 1 ≤ i ≤ n.

The reflection σ_i through the hyperplane perpendicular to α_i is the same as permutation of the adjacent ith and i+1th coordinates. Such transpositions generate the full permutation group. For adjacent simple roots, σ_i(α_i+1) = α_i+1 + α_i = σ_i+1(α_i) = α_i + α_i+1, that is, reflection is equivalent to adding a multiple of 1; but reflection of a simple root perpendicular to a nonadjacent simple root leaves it unchanged, differing by a multiple of 0.

B_n

Let V=Rⁿ, and let Φ consist of all integer vectors in V of length 1 or √2. The total number of roots is 2n². One choice of simple roots is: α_i = e_i - e_i+1, for 1 ≤ i ≤ n-1 (the above choice of simple roots for A_n-1), and the shorter root α_n = e_n.

The reflection σ_n through the hyperplane perpendicular to the short root α_n is of course simply negation of the nth coordinate. For the long simple root α_n-1, σ_n-1(α_n) = α_n + α_n-1, but for reflection perpendicular to the short root, σ_n(α_n-1) = α_n-1 + 2α_n, a difference by a multiple of 2 instead of 1.

B₁ is isomorphic to A₁ via scaling by √2, and is therefore not a distinct root system.

C_n

Let V=Rⁿ, and let Φ consist of all integer vectors in V of length √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. One choice of simple roots is: α_i = e_i - e_i+1, for 1 ≤ i ≤ n-1 (the above choice of simple roots for A_n-1), and the longer root α_n = 2e_n. The reflection σ_n(α_n-1) = α_n-1 + α_n, but σ_n-1(α_n) = α_n + 2α_n-1.

C₂ is isomorphic to B₂ via scaling by √2 and a 45 degree rotation, and is therefore not a distinct root system.

D_n

Let V=Rⁿ, and let Φ consist of all integer vectors in V of length √2. The total number of roots is 2n(n−1). One choice of simple roots is: α_i = e_i - e_i+1, for 1 ≤ i < n (the above choice of simple roots for A_n-1) plus α_n = e_n + e_n-1.

Reflection through the hyperplane perpendicular to α_n is the same as transposing and negating the adjacent nth and n-1th coordinates. Any simple root and its reflection perpendicular to another simple root differ by a multiple of 0 or 1 of the second root, not by any greater multiple.

D₃ reduces to A₃, and is therefore not a distinct root system.

D₄ has additional symmetry called triality.

E₈, E₇, E₆

The E8 lattice can be defined as explicitly by the set of points Γ₈ ⊂ R⁸ such that:

• all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed), and
• the sum of the eight coordinates is an even integer.

Let the E₈ root system be the set of vectors of length √2 in Γ₈, that is: (α ∈ Z⁸ ∪ (Z+½)⁸: |α|² = ∑α_i² = 2, ∑α_i ∈ 2Z).

Then let E₇ be the intersection of E₈ with the hyperplane of vectors perpendicular to a fixed root in E₈, and let E₆ the intersection of E₇ with the hyperplane of vectors perpendicular to a fixed root in E₇. The root systems E₆, E₇, and E₈ have 72, 126, and 240 roots respectively. If we continue to delete roots and reduce dimension, E₅ reduces to D₅, and E₄ reduces to A₄, so no more distinct root systems are found.

An alternative description of the E₈ lattice which is sometimes convenient is the set of all points in Γ′₈ ⊂ R⁸ such that

• all the coordinates are integers and the sum of the coordinates is even, or
• all the coordinates are half-integers and the sum of the coordinates is odd.

The lattices Γ₈ and Γ′₈ are isomorphic and one may pass from one to the other by changing the signs of any odd number of coordinates. The lattice Γ₈ is sometimes called the even coordinate system for E₈ while the lattice Γ′₈ is called the odd coordinate system.

One choice of simple roots for E₈ in the even coordinate system is: α_i = e_i - e_i+1, for 1 ≤ i ≤ 6 and α₇ = e₇ + e₆ (the above choice of simple roots for D₇) along with α₈ = β₀ = = (½,½,½,½,½,½,½,½).

One choice of simple roots for E₈ in the odd coordinate system is: α_i = e_i - e_i+1, for 1 ≤ i ≤ 7 (the above choice of simple roots for A₇) along with α₈ = β₅, where β_j = . (Using β₃ would give an isomorphic result. Using β_1,7 or β_2,6 would simply give A₈ or D₈. As for β₄, its coordinates sum to 0, and the same is true for α_1...7, so they span only the 7-dimensional subspace for which the coordinates sum to 0; in fact -2β₄ = has coordinates (1,2,3,4,3,2,1) in the basis (α_i).)

Deleting α₁ and then α₂ gives sets of simple roots for E₇ and E₆. Since perpendicularity to α₁ means that the first two coordinates are equal, E₇ is then the subspace of E₈ where the first two coordinates are equal, and similarly E₆ is the subspace of E₈ where the first three coordinates are equal. This facilitates explicit definitions of E₇ and E₆ as:

E₇ = (α ∈ Z⁷ ∪ (Z+½)⁷: ∑α_i² + α₁² = 2, ∑α_i + α₁ ∈ 2Z), E₆ = (α ∈ Z⁶ ∪ (Z+½)⁶: ∑α_i² + 2α₁² = 2, ∑α_i + 2α₁ ∈ 2Z)

F₄

For F₄, let V=R⁴, 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. One choice of simple roots is: the choice of simple roots given above for B₃, plus α₄ = .

G₂

G₂ has 12 roots, which form the vertices of a hexagram. See the picture above.

One choice of simple roots is: (α₁, β=α₂-α₁) where α_i = e_i - e_i+1 for i = 1, 2 is the above choice of simple roots for A₂.

Root systems and Lie theory

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

• Simple complex Lie algebras
• 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 weights of the adjoint representation.

In the case of a simply connected connected simple compact Lie group G with maximal torus T, the root lattice can naturally be identified with Hom(T, T) and the coroot lattice with Hom(T, T); see Adams (1983).

Extended and affine Dynkin diagrams

For more details on this topic, see Coxeter group#Affine_Weyl_groups.

There are extensions of Dynkin diagrams, namely extended Dynkin diagrams and affine Dynkin diagrams.

Extended Dynkin diagrams are denoted with a tilde, as in .

Affine Dynkin diagrams describe Cartan matrices of affine Lie algebras.

• Dynkin diagram generator

See also

• Weyl group, Coxeter group
• Coxeter matrix
• ADE classification
• root datum
• Coxeter-Dynkin diagram

References

• Adams, J.F. (1983), Lectures on Lie groups, University of Chicago Press, ISBN 0226005305 
• Bourbaki, Nicolas (2002), Lie groups and Lie algebras, Chapters 4-6 (translated from the 1968 French original by Andrew Pressley), Elements of Mathematics, Springer-Verlag, ISBN 3-540-42650-7 . The classic reference for root systems.

Further reading

• Dynkin, E. B. The structure of semi-simple algebras. (Russian) Uspehi Matem. Nauk (N.S.) 2, (1947). no. 4(20), 59--127.

External links

• John Baez on the ubiquity of Dynkin diagrams in mathematics
• Web tool for making publication-quality Dynkin diagrams with labels (written in JavaScript)