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 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. Finally, root systems are important for their own sake, as in spectral graph theory.

Definitions and 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. These six vectors satisfy the following definition, and therefore they form a root system; this one is known as A₂.

Definition

Let E be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by. A root system in E is a finite set of non-zero vectors that satisfy the following conditions:

The roots span E.
The only scalar multiples of a root that belong to are itself and.
For every root, the set is closed under reflection through the hyperplane perpendicular to.
If and are roots in, then the projection of onto the line through is an integer or half-integer multiple of.

Equivalent ways of writing conditions 3 and 4, respectively, are as follows:

For any two roots, the set contains the element
For any two roots, the number is an integer.

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 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.

Image:Root system A1xA1.svg\|class=skin-invert-image\|150px\|Root system A₁ + A₁	Image:Root system D2.svg\|class=skin-invert-image\|150px\|Root system D₂
Root system	Root system
Image:Root system A2.svg\|class=skin-invert-image\|150px\|Root system A₂	Image:Root system G2.svg\|class=skin-invert-image\|150px\|Root system G₂
Root system	Root system
Image:Root system B2.svg\|class=skin-invert-image\|150px\|Root system B₂	Image:Root system C2.svg\|class=skin-invert-image\|150px\|Root system C₂
Root system	Root system

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

Weyl group

The group of isometries of E 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 reflection planes are the hyperplanes perpendicular to the roots, indicated for by dashed lines in the figure below. The Weyl group is the symmetry group of an equilateral triangle, which has six elements. In this case, the Weyl group is not the full symmetry group of the root system.

Rank one example

There is only one root system of rank 1, consisting of two nonzero vectors. This root system is called.

Rank two examples

In rank 2 there are four possibilities, corresponding to, where. The figure at right shows these possibilities, but with some redundancies: is isomorphic to and is isomorphic to.
Note that a root system is not determined by the lattice that it generates: and both generate a square lattice while and both generate a hexagonal lattice.
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.

Root systems arising from semisimple Lie algebras

If is a complex semisimple Lie algebra and is a Cartan subalgebra, we can construct a root system as follows. We say that is a root of relative to if and there exists some such that
for all. One can show that there is an inner product for which the set of roots forms a root system. The root system of is a fundamental tool for analyzing the structure of and classifying its representations.

History

The concept of a root system was originally introduced by Wilhelm Killing around 1889. He used them in his attempt to classify all simple Lie algebras over the field of complex numbers.
Killing investigated the structure of a Lie algebra by considering what is now called a Cartan subalgebra. Then he studied the roots of the characteristic polynomial, where. Here a root is considered as a function of, or indeed as an element of the dual vector space. This set of roots forms a root system inside, 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 one-half of the square root of a positive integer. This is because and are both integers, by assumption, and
Since, the only possible values for are and, 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. The diagram at right shows that an angle of 60° or 120° corresponds to roots of equal length, while an angle of 45° or 135° corresponds to a length ratio of and an angle of 30° or 150° corresponds to a length ratio of.
In summary, here are the only possibilities for each pair of roots.

Angle of 90 degrees; in that case, the length ratio is unrestricted.
Angle of 60 or 120 degrees, with a length ratio of 1.
Angle of 45 or 135 degrees, with a length ratio of.
Angle of 30 or 150 degrees, with a length ratio of.
Positive roots and simple roots

Given a root system we can always choose a set of positive roots. This is a subset of such that

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

If a set of positive roots is chosen, elements of are called negative roots. A set of positive roots may be constructed by choosing a hyperplane not containing any root and setting to be all the roots lying on a fixed side of. Furthermore, every set of positive roots arises in this way.
An element of is called a simple root if it cannot be written as the sum of two elements of. The set of simple roots is a basis of with the following additional special properties:

Every root is a linear combination of elements of with integer coefficients.
For each, the coefficients in the previous point are either all non-negative or all non-positive.

For each root system there are many different choices of the set of positive roots—or, equivalently, of the simple roots—but any two sets of positive roots differ by the action of the Weyl group.

Dual root system, coroots, and integral elements

The dual root system

If Φ is a root system in E, the coroot α^∨ of a root α is defined by
The set of coroots also forms a root system Φ^∨ in E, called the dual 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^∨.
In the classification described below, the root systems of type and along with the exceptional root systems are all self-dual, meaning that the dual root system is isomorphic to the original root system. By contrast, the and root systems are dual to one another, but not isomorphic.