Root system

Definitions and first examples

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 + nβ, 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.

Definition

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 conditions:

An equivalent way of writing conditions 3 and 4 is as follows:

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

⟨ ⋅ , ⋅ ⟩ : Φ × Φ → Z

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

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 to the right, is said to be irreducible.

Two root systems (E₁, Φ₁) and (E₂, Φ₂) are called isomorphic if there is an invertible linear transformation E₁ → E₂ which sends Φ₁ to Φ₂ such that for each pair of roots, the number ⟨ x , y ⟩ is preserved.

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 two examples

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

In rank 2 there are four possibilities, corresponding to σ α ( β ) = β + n α , where n = 0 , 1 , 2 , 3 . Note that a root system is not determined by the lattice that it generates: A 1 × A 1 and B 2 both generate a square lattice while A 2 and G 2 generate a hexagonal lattice, only two of the five possible types of lattices in two dimensions.

Whenever Φ is a root system in V, and U is a subspace of V spanned by Ψ = Φ ∩ U, then Ψ is a root system in U. 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.

History

The concept of a root system was originally introduced by Wilhelm Killing around 1889 (in German, Wurzelsystem). 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.

Killing investigated the structure of a Lie algebra L , by considering (what is now called) a Cartan subalgebra h . Then he studied the roots of the characteristic polynomial det ( a d L x − t ) , where x ∈ h . Here a root is considered as a function of h , or indeed as an element of the dual vector space h ∗ . This set of roots form a root system inside h ∗ , as defined above, where the inner product is the Killing form.

Elementary consequences of the root system axioms

The cosine of the angle between two roots is constrained to be a half-integral multiple of a square root of an integer. This is because ⟨ β , α ⟩ and ⟨ α , β ⟩ are both integers, by assumption, and

⟨ β , α ⟩ ⟨ α , β ⟩ = 2 ( α , β ) ( α , α ) ⋅ 2 ( α , β ) ( β , β ) = 4 ( α , β ) 2 | α | 2 | β | 2 = 4 cos 2 ⁡ ( θ ) = ( 2 cos ⁡ ( θ ) ) 2 ∈ Z .

Since 2 cos ⁡ ( θ ) ∈ [ − 2 , 2 ] , the only possible values for cos ⁡ ( θ ) are 0 , ± 1 2 , ± 2 2 , ± 3 2 and ± 4 2 = ± 1 , corresponding to angles of 90°, 60° or 120°, 45° or 135°, 30° or 150°, and 0° or 180°. Condition 2 says that no scalar multiples of α other than 1 and -1 can be roots, so 0 or 180°, which would correspond to 2α or −2α, are out.

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 Φ + of Φ such that

For each root α ∈ Φ exactly one of the roots α , – α is contained in Φ + .

For any two distinct α , β ∈ Φ + such that α + β is a root, α + β ∈ Φ + .

If a set of positive roots Φ + is chosen, elements of − Φ + are called negative roots.

An element of Φ + is called a simple root if it cannot be written as the sum of two elements of Φ + . The set Δ of simple roots is a basis of V with the property that every vector in Φ is a linear combination of elements of Δ with all coefficients non-negative, or all coefficients non-positive. For each choice of positive roots, the corresponding set of simple roots is the unique set of roots such that the positive roots are exactly those that can be expressed as a combination of them with non-negative coefficients, and such that these combinations are unique.

The root poset

The set of positive roots is naturally ordered by saying that α ≤ β if and only if β − α is a nonnegative linear combination of simple roots. This poset is graded by deg ⁡ ( ∑ α ∈ Δ λ α α ) = ∑ α ∈ Δ λ α , and has many remarkable combinatorial properties, one of them being that one can determine the degrees of the fundamental invariants of the corresponding Weyl group from this poset. The Hasse graph is a visualization of the ordering of the root poset.

Dual root system and coroots

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

α ∨ = 2 ( α , α ) α .

The set of coroots also forms a root system Φ^∨ in V, 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 V spanned by Φ^∨ is called the coroot lattice. Both Φ and Φ^∨ have the same Weyl group W and, for s in W,

( s α ) ∨ = s ( α ∨ ) .

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

Classification of root systems by Dynkin diagrams

A root system is irreducible if it can not be partitioned into the union of two proper subsets Φ = Φ 1 ∪ Φ 2 , such that ( α , β ) = 0 for all α ∈ Φ 1 and β ∈ Φ 2 .

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 2 π / 3 radians, a directed double edge if they make an angle of 3 π / 4 radians, and a directed triple edge if they make an angle of 5 π / 6 radians. 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. Root systems are irreducible if and only if their Dynkin diagrams are connected. 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

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 at most two values for the length (α, α)^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.

An

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 i-th and (i + 1)-th 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.

The A_n root lattice - that is, the lattice generated by the A_n roots - is most easily described as the set of integer vectors in Rⁿ⁺¹ whose components sum to zero.

The A₃ root lattice is known to crystallographers as the face-centered cubic (fcc) (or cubic close packed) lattice.

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

Bn

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.

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

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

Cn

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.

The C_n root lattice - that is, the lattice generated by the C_n 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.

Root system B₃, C₃, and A₃=D₃ as points within a cube and octahedron

Dn

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-1 (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 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 D_n root lattice - that is, the lattice generated by the D_n roots - consists of all integer vectors whose components sum to an even integer. This is the same as the C_n root lattice.

D₃ coincides with A₃, and is therefore not a distinct root system.

D₄ has additional symmetry called triality.

E6, E7, E8

The E₈ root system is any set of vectors in R⁸ that is congruent to the following set:

D₈ ∪ { ½( ∑_i=1⁸ ε_ie_i) : ε_i = ±1, ε₁•••ε₈ = +1}.

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:

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

Thus,

E₈ = {α ∈ Z⁸ ∪ (Z+½)⁸ : |α|² = ∑α_i² = 2, ∑α_i ∈ 2Z}.

The root system E₇ is the set of vectors in E₈ that are perpendicular to a fixed root in E₈. The root system E₇ has 126 roots.

The root system E₆ is not the set of vectors in E₇ that are perpendicular to a fixed root in E₇, indeed, one obtains D₆ that way. However, E₆ is the subsystem of E₈ perpendicular to two suitably chosen roots of E₈. The root system E₆ has 72 roots.

An alternative description of the E₈ lattice which is sometimes convenient is as the set Γ'₈ of all points in 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; 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:

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 = − 1 2 ( ∑ i = 1 8 e i ) = (-½,-½,-½,-½,-½,-½,-½,-½).

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:

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 = 1 2 ( − ∑ i = 1 j e i + ∑ i = j + 1 8 e i ) .

(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).)

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:

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}

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.

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 = – 1 2 ∑ i = 1 4 e i .

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.

G2

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 = α₂ – α₁) where α_i = e_i – e_i+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.

Root systems and Lie theory

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

simple Lie groups (see the list of simple Lie groups), including the

simple complex Lie groups;

their associated simple complex Lie algebras; and

simply connected complex Lie groups which are simple modulo centers.

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

In the case of a simply 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), where T is the circle group; see Adams (1983).

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