# Root system

*See also Simple Lie group.*

In mathematics, a **root system** is a kind of configuration in Euclidean space that has turned out to be fundamental in Lie group theory. Since Lie groups (and some analogues such as algebraic groups) became used in most 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.

Table of contents |

2 Classification 3 Dynkin diagrams 4 A _{n}5 B _{n}6 C _{n}7 D _{n}8 E _{n}9 F _{4}10 G _{2}11 Root systems and Lie theory |

## Definitions

Formally, a **root system** is a finite set Φ of non-zero vectors (**roots**) spanning a finite-dimensional Euclidean space *V* which satisfy the following properties:

- The only scalar multiples of a root α in
*V*which belong to*Φ*are α itself and −α. - For every root α in
*V*, the set Φ is symmetric under reflection through the hyperplane of vectors perpendicular to α - If α and β are vectors in Φ, the projection of 2β onto the line through α is an integer multiple of α

**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 A

_{2}, B

_{2}, and G

_{2}pictured below, is said to be

**irreducible**.

Two irreducible root systems (*E*_{1},Φ_{1}) and (*E*_{2},Φ_{2}) are considered to be the same if there is an invertible linear transformation *E*_{1}→*E*_{2} which preserves distance up to a scale factor and which sends Φ_{1} to Φ_{2}.

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.

## Classification

It is not too difficult to classify the root systems of rank 2:

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 in a root system. In particular, two such roots meet at an angle of 0, 30, 45, 60, 90, 120, 135, 150, or 180 degrees.

In general, irreducible root systems are specified by a family (indicated by a letter A to G) and the rank (indicated by a subscript). There are four infinite families and five exceptional cases:

## Dynkin diagrams

To prove this classification theorem, one uses the angles between pairs of roots to encode the root system in a much simpler combinatorial object, the **Dynkin diagram**. The Dynkin diagrams can then be classified according to the scheme given above.

To every root system is associated a graph (possibly with a specially marked edge) called the **Dynkin diagram**. The Dynkin diagram can be extracted from the root system by choosing a **base**, that is a subset Δ of Φ which is a basis of *V* with the special property that every vector in Φ when written in the basis Δ has either all coefficients ≥0 or else all ≤0.

The vertices of the Dynkin diagram correspond to vectors in Δ. An edge is drawn between each non-orthogonal pair of vectors; it is a double edge if they make an angle of 135 degrees, and a triple edge if they make an angle of 150 degrees. In addition, double and triple edges are marked with an angle sign pointing toward the shorter vector.

Although a given root system has more than one base, the Weyl group acts transitively on the set of bases. Therefore, the root system determines the Dynkin diagram. Given two root systems with the same Dynkin diagram, we 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, and 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:

In detail, the individual root systems can be realized case-by-case, as in the following sections.

## A_{n}

## B_{n}

## C_{n}

## D_{n}

## E_{n}

For *V*_{8}, let *V*=**R**^{8}, and let E_{8} denote the set of vectors α of length √2 such that the coordinates of 2α are all integers and are either all even or all odd. Then E_{7} can be constructed as the intersection of E_{8} with the hyperplane of vectors perpendicular to a fixed root α in E_{8}, and E_{6} can be constructed as the intersection of E_{8} with two such hyperplanes corresponding to roots α and β which are neither orthogonal to one another nor scalar multiples of one another. The root systems E_{6}, E_{7}, and E_{8} have 72, 126, and 240 roots respectively.

## F_{4}

For F_{4}, let *V*=**R**^{4}, 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.

## G_{2}

There are 12 roots in G_{2}, which form the vertices of a hexagram. See the picture above.

## 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

See also Weyl group, Coxeter group, Cartan matrix, Coxeter matrix

A

**root system**can also be said to describe a plant's root and associated systems.