Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important 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 theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory.[1]

As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors span the whole space. If you consider the line perpendicular to any root, say β, then the reflection of R2 in that line sends any other root, say α, to another root. Moreover, the root to which it is sent equals α + , where n is an integer (in this case, n equals 1). These six vectors satisfy the following definition, and therefore they form a root system; this one is known as A2.

Some authors only include conditions 1–3 in the definition of a root system.[4] In this context, a root system that also satisfies the integrality condition is known as a crystallographic root system.[5] Other authors omit condition 2; then they call root systems satisfying condition 2 reduced.[6] 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 σα(β) differ by an integer multiple of α. Note that the operator

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 A2, B2, and G2 pictured to the right, is said to be irreducible.

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

Whenever Φ is a root system in E, and S is a subspace of E spanned by Ψ = Φ ∩ S, then Ψ is a root system in S. Thus, the exhaustive list of four 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 must meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees.

The concept of a root system was originally introduced by Wilhelm Killing around 1889 (in German, Wurzelsystem[10]).[11] He used them in his attempt to classify all simple Lie algebras over the field of complex numbers. Killing originally made a mistake in the classification, listing two exceptional rank 4 root systems, when in fact there is only one, now known as F4. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.[12]

In summary, here are the only possibilities for each pair of roots.[13]

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

If Δ is a set of simple roots for Φ, then Δ is a set of simple roots for Φ.[17]

The set of integral elements is called the weight lattice associated to the given root system. This term comes from the representation theory of semisimple Lie algebras, where the integral elements form the possible weights of finite-dimensional representations.

The definition of a root system guarantees that the roots themselves are integral elements. Thus, every integer linear combination of roots is also integral. In most cases, however, there will be integral elements that are not integer combinations of roots. That is to say, in general the weight lattice does not coincide with the root lattice.

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 the roots in Δ. Edges are drawn between vertices as follows, according to the angles. (Note that the angle between simple roots is always at least 90 degrees.)

The term "directed edge" means that double and triple edges are marked with an arrow pointing toward the shorter vector. (Thinking of the arrow as a "greater than" sign makes it clear which way the arrow is supposed to point.)

Although a given root system has more than one possible set of simple roots, the Weyl group acts transitively on such choices.[19] 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.[20]

Thus the problem of classifying root systems reduces to the problem of classifying possible Dynkin diagrams. A root systems is irreducible if and only if its Dynkin diagrams is connected.[21] The possible connected diagrams are as indicated in the figure. The subscripts indicate the number of vertices in the diagram (and hence the rank of the corresponding irreducible root system).

Theorem: The Weyl group acts freely and transitively on the Weyl chambers. Thus, the order of the Weyl group is equal to the number of Weyl chambers.

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

In each case, the roots are non-zero weights of the adjoint representation.

For connections between the exceptional root systems and their Lie groups and Lie algebras see E8, E7, E6, F4, and G2.

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 an irreducible root system there can be at most two values for the length (αα)1/2, 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 adjacent table, |Φ<| 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 |W| denotes the order of the Weyl group.

Let E be the subspace of Rn+1 for which the coordinates sum to 0, and let Φ be the set of vectors in E 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: αi = eiei+1, for 1 ≤ i ≤ n.

The reflection σi through the hyperplane perpendicular to αi is the same as permutation of the adjacent i-th and (i + 1)-th coordinates. Such transpositions generate the full permutation group. For adjacent simple roots, σi(αi+1) = αi+1 + αiσi+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 An root lattice – that is, the lattice generated by the An roots – is most easily described as the set of integer vectors in Rn+1 whose components sum to zero.

The A2 root lattice is the vertex arrangement of the triangular tiling.

The A3 root lattice is known to crystallographers as the face-centered cubic (or cubic close packed) lattice.[29] It is the vertex arrangement of the tetrahedral-octahedral honeycomb.

The A3 root system (as well as the other rank-three root systems) may be modeled in the Zometool Construction set.[30]

In general, the An root lattice is the vertex arrangement of the n-dimensional simplectic honeycomb.

Let E = Rn, and let Φ consist of all integer vectors in E of length 1 or 2. The total number of roots is 2n2. One choice of simple roots is: αi = eiei+1, for 1 ≤ in – 1 (the above choice of simple roots for An−1), and the shorter root αn = en.

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.

The Bn root lattice – that is, the lattice generated by the Bn roots – consists of all integer vectors.

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

Let E = Rn, and let Φ consist of all integer vectors in E 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: αi = eiei+1, for 1 ≤ in − 1 (the above choice of simple roots for An−1), and the longer root αn = 2en. The reflection σn(αn−1) = αn−1 + αn, but σn−1(αn) = αn + 2αn−1.

The Cn root lattice – that is, the lattice generated by the Cn roots – consists of all integer vectors whose components sum to an even integer.

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

Let E = Rn, and let Φ consist of all integer vectors in E of length 2. The total number of roots is 2n(n − 1). One choice of simple roots is: αi = eiei+1, for 1 ≤ in − 1 (the above choice of simple roots for An−1) plus αn = en + en−1.

Reflection through the hyperplane perpendicular to αn is the same as transposing and negating the adjacent n-th and (n − 1)-th 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.

The Dn root lattice – that is, the lattice generated by the Dn roots – consists of all integer vectors whose components sum to an even integer. This is the same as the Cn root lattice.

The Dn roots are expressed as the vertices of a rectified n-orthoplex, Coxeter-Dynkin diagram: CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.png...CDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png. The 2n(n−1) vertices exist in the middle of the edges of the n-orthoplex.

D3 coincides with A3, and is therefore not a distinct root system. The 12 D3 root vectors are expressed as the vertices of CDel node.pngCDel split1.pngCDel nodes 11.png, a lower symmetry construction of the cuboctahedron.

D4 has additional symmetry called triality. The 24 D4 root vectors are expressed as the vertices of CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, a lower symmetry construction of the 24-cell.

The root system has 240 roots. The set just listed is the set of vectors of length 2 in the E8 root lattice, also known simply as the E8 lattice or Γ8. This is the set of points in R8 such that:

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

The lattices Γ8 and Γ'8 are isomorphic; 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 with rows ordered by node order in the alternate (non-canonical) Dynkin diagrams (above) is:

One choice of simple roots for E8 in the odd coordinate system with rows ordered by node order in alternate (non-canonical) Dynkin diagrams (above) is:

(Using β3 would give an isomorphic result. Using β1,7 or β2,6 would simply give A8 or D8. As for β4, its coordinates sum to 0, and the same is true for α1...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 (αi).)

Since perpendicularity to α1 means that the first two coordinates are equal, E7 is then the subset of E8 where the first two coordinates are equal, and similarly E6 is the subset of E8 where the first three coordinates are equal. This facilitates explicit definitions of E7 and E6 as:

Note that deleting α1 and then α2 gives sets of simple roots for E7 and E6. However, these sets of simple roots are in different E7 and E6 subspaces of E8 than the ones written above, since they are not orthogonal to α1 or α2.

The F4 root lattice – that is, the lattice generated by the F4 root system – is the set of points in R4 such that either all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed). This lattice is isomorphic to the lattice of Hurwitz quaternions.

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

One choice of simple roots is: (α1, β = α2α1) where αi = eiei+1 for i = 1, 2 is the above choice of simple roots for A2.

The G2 root lattice – that is, the lattice generated by the G2 roots – is the same as the A2 root lattice.