# 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, **R**^{2}, 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 **R**^{2} 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 *A*_{2}.

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 *σ*_{α}(*β*) 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.

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 *A*_{2}, *B*_{2}, and *G*_{2} 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 F_{4}. 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 vectors 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 E_{8}, E_{7}, E_{6}, F_{4}, and G_{2}.

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 *r*^{2}/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 **R**^{n+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 **R**^{n+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 *n*^{2} + *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 (

**)-th coordinates. Such transpositions generate the full permutation group. For adjacent simple roots,**

*i*+ 1*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 **R**^{n+1} whose components sum to zero.

The A_{2} root lattice is the vertex arrangement of the triangular tiling.

The A_{3} 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 A_{3} root system (as well as the other rank-three root systems) may be modeled in the Zometool Construction set.^{[30]}

In general, the A_{n} root lattice is the vertex arrangement of the *n*-dimensional simplectic honeycomb.

Let *E* = **R**^{n}, and let Φ consist of all integer vectors in *E* of length 1 or √2. The total number of roots is 2*n*^{2}. 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 *n*th 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} + 2**α**_{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.

*B*_{1} is isomorphic to A_{1} via scaling by √2, and is therefore not a distinct root system.

Let *E* = **R**^{n}, and let Φ consist of all integer vectors in *E* of length √2 together with all vectors of the form 2*λ*, where *λ* is an integer vector of length 1. The total number of roots is 2*n*^{2}. 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} = 2**e**_{n}.
The reflection *σ*_{n}(**α**_{n−1}) = **α**_{n−1} + **α**_{n}, but *σ*_{n−1}(**α**_{n}) = **α**_{n} + 2**α**_{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.

C_{2} is isomorphic to B_{2} via scaling by √2 and a 45 degree rotation, and is therefore not a distinct root system.

Let *E* = **R**^{n}, and let Φ consist of all integer vectors in *E* of length √2. The total number of roots is 2*n*(*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.

The *D*_{n} roots are expressed as the vertices of a rectified *n*-orthoplex, Coxeter-Dynkin diagram: .... The 2*n*(*n*−1) vertices exist in the middle of the edges of the *n*-orthoplex.

D_{3} coincides with A_{3}, and is therefore not a distinct root system. The 12 D_{3} root vectors are expressed as the vertices of , a lower symmetry construction of the cuboctahedron.

D_{4} has additional symmetry called triality. The 24 D_{4} root vectors are expressed as the vertices of , 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 Γ_{8}. This is the set of points in **R**^{8} such that:

An alternative description of the E_{8} lattice which is sometimes convenient is as the set Γ'_{8} of all points in **R**^{8} such that

The lattices Γ_{8} 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 Γ_{8} is sometimes called the *even coordinate system* for E_{8} while the lattice Γ'_{8} is called the *odd coordinate system*.

One choice of simple roots for E_{8} 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 E_{8} in the odd coordinate system with rows ordered by node order in alternate (non-canonical) Dynkin diagrams (above) is:

(Using **β**_{3} would give an isomorphic result. Using **β**_{1,7} or **β**_{2,6} would simply give A_{8} or D_{8}. As for **β**_{4}, 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 –2**β**_{4} has coordinates (1,2,3,4,3,2,1) in the basis (**α**_{i}).)

Since perpendicularity to **α**_{1} means that the first two coordinates are equal, E_{7} is then the subset of E_{8} where the first two coordinates are equal, and similarly E_{6} is the subset of E_{8} where the first three coordinates are equal. This facilitates explicit definitions of E_{7} and E_{6} as:

Note that deleting **α**_{1} and then **α**_{2} gives sets of simple roots for E_{7} and E_{6}. However, these sets of simple roots are in different E_{7} and E_{6} subspaces of E_{8} than the ones written above, since they are not orthogonal to **α**_{1} or **α**_{2}.

The F_{4} root lattice – that is, the lattice generated by the F_{4} root system – is the set of points in **R**^{4} 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 G_{2} has 12 roots, which form the vertices of a hexagram. See the picture above.

One choice of simple roots is: (**α**_{1}, **β** = **α**_{2} – **α**_{1}) where **α**_{i} = **e**_{i} – **e**_{i+1} for *i* = 1, 2 is the above choice of simple roots for *A*_{2}.

The *G*_{2} root lattice – that is, the lattice generated by the *G*_{2} roots – is the same as the *A*_{2} root lattice.