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).
In view of property 3, the integrality condition is equivalent to stating that β and its reflection σα(β) differ by an integer multiple of α. Note that the operator
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.
In rank 2 there are four possibilities, corresponding to σα(β) = β + nα, where n = 0, 1, 2, 3.
|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.
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.
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).
|Bn (n≥2)||2n2||2n||2||2n n!|
|Cn (n≥3)||2n2||2n(n−1)||2||2n n!|
|Dn (n≥4)||2n(n−1)||4||2n−1 n!|
Irreducible root systems are named according to their corresponding connected Dynkin diagrams. There are four infinite families (An, Bn, Cn, and Dn, called the classical root systems) and five exceptional cases (the exceptional root systems). The subscript indicates the rank of the root system. In the table to the right, denotes the number of short roots (if all roots have the same length they are taken to be long by definition), I denotes the determinant of the Cartan matrix, and denotes the order of the Weyl group.
The reflection σi through the hyperplane perpendicular to αi is the same as permutation of the adjacent ith and i+1th coordinates. Such transpositions generate the full permutation group. For adjacent simple roots, σi(αi+1) = αi+1 + αi = σi+1(αi) = αi + αi+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.
The reflection σn through the hyperplane perpendicular to the short root αn is of course simply negation of the nth coordinate. For the long simple root αn-1, σn-1(αn) = αn + αn-1, but for reflection perpendicular to the short root, σn(αn-1) = αn-1 + 2αn, 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.
C2 is isomorphic to B2 via scaling by √2 and a 45 degree rotation, and is therefore not a distinct root system.
Reflection through the hyperplane perpendicular to αn 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.
Let the E8 root system be the set of vectors of length √2 in Γ8, that is: (α ∈ Z8 ∪ (Z+½)8: |α|2 = ∑αi2 = 2, ∑αi ∈ 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.
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: αi = ei - ei+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).
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:
One choice of simple roots is: (α1, β=α2-α1) where αi = ei - ei+1 for i = 1, 2 is the above choice of simple roots for A2.
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.