Lie groups and Lie algebras |
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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, 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}.
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 (called roots) that satisfy the following conditions:^{[2]}^{[3]}
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.^{[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
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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.
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.^{[7]}
The root lattice of a root system Φ is the Z-submodule of E generated by Φ. It is a lattice in E.
Main article: 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 (e.g., a 60-degree rotation is a symmetry of the root system but not an element of the Weyl group).
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 .^{[8]} 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.
See also: Semisimple Lie algebra § Cartan subalgebras and root systems, and Root system of a semi-simple Lie algebra |
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
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]}
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.^{[11]}
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.^{[13]}
Given a root system we can always choose (in many ways) a set of positive roots. This is a subset of such that
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.^{[14]}
An element of is called a simple root (also fundamental root) if it cannot be written as the sum of two elements of . (The set of simple roots is also referred to as a base for .) The set of simple roots is a basis of with the following additional special properties:^{[15]}
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.^{[16]}
See also: Langlands dual group |
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 (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]}
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 (except when ).
See also: Weight (representation theory) § Weights in the representation theory of semisimple Lie algebras |
A vector in E is called integral^{[18]} if its inner product with each coroot is an integer:
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.
See also: Dynkin diagram |
A root system is irreducible if it cannot be partitioned into the union of two proper subsets , such that for all and .
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 vertices 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.)
Note that by the elementary properties of roots noted above, the rules for creating the Dynkin diagram can also be described as follows. No edge if the roots are orthogonal; for nonorthogonal roots, a single, double, or triple edge according to whether the length ratio of the longer to shorter is 1, , . In the case of the root system for example, there are two simple roots at an angle of 150 degrees (with a length ratio of ). Thus, the Dynkin diagram has two vertices joined by a triple edge, with an arrow pointing from the vertex associated to the longer root to the other vertex. (In this case, the arrow is a bit redundant, since the diagram is equivalent whichever way the arrow goes.)
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).
If is a root system, the Dynkin diagram for the dual root system is obtained from the Dynkin diagram of by keeping all the same vertices and edges, but reversing the directions of all arrows. Thus, we can see from their Dynkin diagrams that and are dual to each other.
See also: Coxeter group § Affine Coxeter groups |
If is a root system, we may consider the hyperplane perpendicular to each root . Recall that denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of generated by all the 's. The complement of the set of hyperplanes is disconnected, and each connected component is called a Weyl chamber. If we have fixed a particular set Δ of simple roots, we may define the fundamental Weyl chamber associated to Δ as the set of points such that for all .
Since the reflections preserve , they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers.
The figure illustrates the case of the root system. The "hyperplanes" (in this case, one dimensional) orthogonal to the roots are indicated by dashed lines. The six 60-degree sectors are the Weyl chambers and the shaded region is the fundamental Weyl chamber associated to the indicated base.
A basic general theorem about Weyl chambers is this:^{[22]}
In the case, for example, the Weyl group has six elements and there are six Weyl chambers.
A related result is this one:^{[23]}
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.
We now give a brief indication of how irreducible root systems classify simple Lie algebras over , following the arguments in Humphreys.^{[24]} A preliminary result says that a semisimple Lie algebra is simple if and only if the associated root system is irreducible.^{[25]} We thus restrict attention to irreducible root systems and simple Lie algebras.
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}.
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^{n−1} | 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). 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.
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 |
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 ith 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_{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.
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 |
α_{4} | 0 | 0 | 0 | 1 |
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 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.
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 |
α_{4} | 0 | 0 | 0 | 2 |
Let E = R^{n}, and let Φ consist of all integer vectors in E 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.
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 |
α_{4} | 0 | 0 | 1 | 1 |
Let E = R^{n}, and let Φ consist of all integer vectors in E 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}) together with α_{n} = e_{n−1} + e_{n}.
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 2n(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 twelve 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 twenty-four D_{4} root vectors are expressed as the vertices of , a lower symmetry construction of the 24-cell.
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 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:
Thus,
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 |
−1/2 | −1/2 | −1/2 | −1/2 | −1/2 | −1/2 | −1/2 | −1/2 |
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:
(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 |
−1/2 | −1/2 | −1/2 | −1/2 | −1/2 | 1/2 | 1/2 | 1/2 |
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
(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}).)
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}.
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 | 0 |
α_{4} | −1/2 | −1/2 | −1/2 | −1/2 |
For F_{4}, let E = 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 .
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.
e_{1} | e_{2} | e_{3} | |
---|---|---|---|
α_{1} | 1 | −1 | 0 |
β | −1 | 2 | −1 |
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.
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.^{[31]} The Hasse graph is a visualization of the ordering of the root poset.