Root system
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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. 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 graph theory in the study of eigenvalues.
Definitions and first examples
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+ n β , 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}.
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:^{[1]}^{[2]} Template:Ordered list 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.^{[3]} In this context, a root system that also satisfies the integrality condition is known as a crystallographic root system.^{[4]} Other authors omit condition 2; then they call root systems satisfying condition 2 reduced.^{[5]} 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.
Root system Template:Dynkin |
Root system Template:Dynkin |
Root system Template:Dynkin |
Root system Template:Dynkin |
Root system Template:Dynkin |
Root system Template:Dynkin |
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 A_{2}, B_{2}, and G_{2} pictured to the right, is said to be irreducible.
Two root systems (E_{1}, Φ_{1}) and (E_{2}, Φ_{2}) are called isomorphic if there is an invertible linear transformation E_{1} → E_{2} which sends Φ_{1} to Φ_{2} such that for each pair of roots, the number is preserved.^{[6]}
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 Template:Visible anchor 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 . This root system is called .
In rank 2 there are four possibilities, corresponding to , where . Note that a root system that generates a lattice is not unique: and generate a square lattice while and 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^{[7]}).^{[8]} 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.^{[9]}
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 form a root system inside , as defined above, where the inner product is the Killing form.^{[10]}
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 and are both integers, by assumption, and
Since , the only possible values for are , 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 of Φ 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.
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 property that every vector in is a linear combination of elements of 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 if and only if is a nonnegative linear combination of simple roots. This poset is graded by , 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.^{[11]} The Hasse graph is a visualization of the ordering of the root poset.
Dual root system and coroots
{{#invoke:see also|seealso}} If Φ is a root system in V, the coroot α^{V} of a root α is defined by
The set of coroots also forms a root system Φ^{V} in V, called the dual root system (or sometimes inverse root system). By definition, α^{V V} = α, so that Φ is the dual root system of Φ^{V}. The lattice in V spanned by Φ^{V} is called the coroot lattice. Both Φ and Φ^{V} have the same Weyl group W and, for s in W,
If Δ is a set of simple roots for Φ, then Δ^{V} is a set of simple roots for Φ^{V}.
Classification of root systems by Dynkin diagrams
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 radians, a directed double edge if they make an angle of radians, and a directed triple edge if they make an angle of 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
I | D | ||||
---|---|---|---|---|---|
A_{n} (n ≥ 1) | n(n + 1) | n + 1 | (n + 1)! | ||
B_{n} (n ≥ 2) | 2n^{2} | 2n | 2 | 2 | 2^{n} n! |
C_{n} (n ≥ 3) | 2n^{2} | 2n(n − 1) | 2 | 2 | 2^{n} n! |
D_{n} (n ≥ 4) | 2n(n − 1) | 4 | 2^{n − 1} n! | ||
E_{6} | 72 | 3 | 51840 | ||
E_{7} | 126 | 2 | 2903040 | ||
E_{8} | 240 | 1 | 696729600 | ||
F_{4} | 48 | 24 | 4 | 1 | 1152 |
G_{2} | 12 | 6 | 3 | 1 | 12 |
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).^{[12]} 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 table to the right, |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.
Explicit construction of the irreducible root systems
A_{n}
e_{1} | e_{2} | e_{3} | e_{4} | |
---|---|---|---|---|
α_{1} | 1 | −1 | 0 | 0 |
α_{2} | 0 | 1 | −1 | 0 |
α_{3} | 0 | 0 | 1 | −1 |
Template:Dynkin2 |
Let V be the subspace of R^{n+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 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 (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 R^{n+1} whose components sum to zero.
The A_{3} root lattice is known to crystallographers as the face-centered cubic (fcc) (or cubic close packed) lattice.^{[13]}
B_{n}
1 | -1 | 0 | 0 |
0 | 1 | -1 | 0 |
0 | 0 | 1 | -1 |
0 | 0 | 0 | 1 |
Template:Dynkin2 |
Let V = R^{n}, and let Φ consist of all integer vectors in V of length 1 or √2. The total number of roots is 2n^{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 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.
B_{1} is isomorphic to A_{1} via scaling by √2, and is therefore not a distinct root system.
C_{n}
1 | -1 | 0 | 0 |
0 | 1 | -1 | 0 |
0 | 0 | 1 | -1 |
0 | 0 | 0 | 2 |
Template:Dynkin2 |
Let V = R^{n}, 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 2n^{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} = 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.
C_{2} is isomorphic to B_{2} via scaling by √2 and a 45 degree rotation, and is therefore not a distinct root system.
Root system B_{3}, C_{3}, and A_{3}=D_{3} as points within a cube and octahedron
D_{n}
1 | -1 | 0 | 0 |
0 | 1 | -1 | 0 |
0 | 0 | 1 | -1 |
0 | 0 | 1 | 1 |
Let V = R^{n}, 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 (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.
D_{3} reduces to A_{3}, and is therefore not a distinct root system.
D_{4} has additional symmetry called triality.
E_{6}, E_{7}, E_{8}
72 vertices of 1_{22} represent the root vectors of E_{6} (Green nodes are doubled in this E6 Coxeter plane projection) |
126 vertices of 2_{31} represent the root vectors of E_{7} |
240 vertices of 4_{21} represent the root vectors of E_{8} |
- The E_{8} root system is any set of vectors in R^{8} that is congruent to the following set:
- D_{8} ∪ { ½( ∑_{i=1}^{8} ε_{i}e_{i}) : ε_{i} = ±1, ε_{1}•••ε_{8} = +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 Γ_{8}. This is the set of points in R^{8} such that:
- all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed), and
- the sum of the eight coordinates is an even integer.
Thus,
- E_{8} = {α ∈ Z^{8} ∪ (Z+½)^{8} : |α|^{2} = ∑α_{i}^{2} = 2, ∑α_{i} ∈ 2Z}.
- The root system E_{7} is the set of vectors in E_{8} that are perpendicular to a fixed root in E_{8}. The root system E_{7} has 126 roots.
- The root system E_{6} is not the set of vectors in E_{7} that are perpendicular to a fixed root in E_{7}, indeed, one obtains D_{6} that way. However, E_{6} is the subsystem of E_{8} perpendicular to two suitably chosen roots of E_{8}. The root system E_{6} has 72 roots.
1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | -1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | -1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | -1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
-½ | -½ | -½ | -½ | -½ | -½ | -½ | -½ |
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
- 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 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:
- α_{i} = e_{i} – e_{i+1}, for 1 ≤ i ≤ 6, and
- α_{7} = e_{7} + e_{6}
(the above choice of simple roots for D_{7}) along with
1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | -1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | -1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | -1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 |
-½ | -½ | -½ | -½ | -½ | ½ | ½ | ½ |
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:
- α_{i} = e_{i} – e_{i+1}, for 1 ≤ i ≤ 7
(the above choice of simple roots for A_{7}) along with
(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 –2β_{4} has coordinates (1,2,3,4,3,2,1) in the basis (α_{i}).)
Deleting α_{1} and then α_{2} gives sets of simple roots for E_{7} and E_{6}. 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:
- E_{7} = {α ∈ Z^{7} ∪ (Z+½)^{7} : ∑α_{i}^{2} + α_{1}^{2} = 2, ∑α_{i} + α_{1} ∈ 2Z},
- E_{6} = {α ∈ Z^{6} ∪ (Z+½)^{6} : ∑α_{i}^{2} + 2α_{1}^{2} = 2, ∑α_{i} + 2α_{1} ∈ 2Z}
F_{4}
1 | -1 | 0 | 0 |
0 | 1 | -1 | 0 |
0 | 0 | 1 | 0 |
-½ | -½ | -½ | -½ |
Template:Dynkin2 |
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. One choice of simple roots is: the choice of simple roots given above for B_{3}, plus α_{4} = – .
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.
G_{2}
1 | -1 | 0 |
-1 | 2 | -1 |
Template:Dynkin2 |
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.
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 Template:Harvtxt.
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}.
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See also
- ADE classification
- Affine root system
- Coxeter–Dynkin diagram
- Coxeter group
- Coxeter matrix
- Dynkin diagram
- root datum
- Root system of a semi-simple Lie algebra
- Weyl group
Notes
- ↑ Bourbaki, Ch.VI, Section 1
- ↑ Humphreys (1972), p.42
- ↑ Humphreys (1992), p.6
- ↑ Humphreys (1992), p.39
- ↑ Humphreys (1992), p.41
- ↑ Humphreys (1972), p.43
- ↑ Killing (1889)
- ↑ Bourbaki (1998), p.270
- ↑ Coleman, p.34
- ↑ Bourbaki (1998), p.270
- ↑ Humphreys (1992), Theorem 3.20
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}.
- ↑ Conway, John Horton; Sloane, Neil James Alexander; & Bannai, Eiichi. Sphere packings, lattices, and groups. Springer, 1999, Section 6.3.
References
- {{#invoke:citation/CS1|citation
|CitationClass=citation }}
- {{#invoke:citation/CS1|citation
|CitationClass=citation }}. The classic reference for root systems.
- Template:Cite isbn
- {{#invoke:citation/CS1|citation
|CitationClass=citation }}
- Template:Cite isbn
- Template:Cite isbn
- Killing, Die Zusammensetzung der stetigen endlichen Transformationsgruppen Mathematische Annalen, Part 1: Volume 31, Number 2 June 1888, Pages 252-290 Template:Hide in printTemplate:Only in print; Part 2: Volume 33, Number 1 March 1888, Pages 1–48 Template:Hide in printTemplate:Only in print; Part3: Volume 34, Number 1 March 1889, Pages 57–122 Template:Hide in printTemplate:Only in print; Part 4: Volume 36, Number 2 June 1890,Pages 161-189 Template:Hide in printTemplate:Only in print
- {{#invoke:citation/CS1|citation
|CitationClass=citation }}.
Further reading
- Dynkin, E. B. The structure of semi-simple algebras.
- REDIRECT Template:Link language Uspehi Matem. Nauk (N.S.) 2, (1947). no. 4(20), 59–127.