Root system

 ( Euclidean Geometry ) 

In mathematics, a root system is a configuration of vector space in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of and , especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as ) 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 , 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.

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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 Linear 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 (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 A₂.

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Definition Let E be a finite-dimensional Euclidean space vector space, with the standard Dot product denoted by $(\backslash cdot,\backslash cdot)$. A root system $\backslash Phi$ in E is a finite set of non-zero vectors (called roots) that satisfy the following conditions:Bourbaki, Ch.VI, Section 1

1. The roots linear span E.
2. The only scalar multiples of a root $\backslash alpha\backslash in\backslash Phi$ that belong to $\backslash Phi$ are $\backslash alpha$ itself and $-\backslash alpha$.
3. For every root $\backslash alpha\backslash in\backslash Phi$, the set $\backslash Phi$ is closed under reflection through the hyperplane perpendicular to $\backslash alpha$.
4. ( Integrality) If $\backslash alpha$ and $\backslash beta$ are roots in $\backslash Phi$, then the projection of $\backslash beta$ onto the line through $\backslash alpha$ is an integer or half-integer multiple of $\backslash alpha$.

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

3. For any two roots $\backslash alpha,\backslash beta\; \backslash in\; \backslash Phi$, the set $\backslash Phi$ contains the element $\backslash sigma\_\backslash alpha(\backslash beta):=\backslash beta-2\backslash frac\{(\backslash alpha,\backslash beta)\}\{(\backslash alpha,\backslash alpha)\}\backslash alpha.$
4. For any two roots $\backslash alpha,\backslash beta\backslash in\backslash Phi$, the number $\backslash langle\; \backslash beta,\; \backslash alpha\; \backslash rangle\; :=\; 2\; \backslash frac\{(\backslash alpha,\backslash beta)\}\{(\backslash alpha,\backslash alpha)\}$ 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( β) differ by an integer multiple of  α. Note that the operator $$\backslash langle\; \backslash cdot,\; \backslash cdot\; \backslash rangle\; \backslash colon\; \backslash Phi\; \backslash times\; \backslash Phi\; \backslash to\; \backslash mathbb\{Z\}$$ defined by property 4 is not an inner product. It is not necessarily symmetric and is linear only in the first argument.

+ Rank-2 root systems
Root system $A\_1\; \backslash times\; A\_1$	Root system $D\_2$
Root system $A\_2$	Root system $G\_2$
Root system $B\_2$	Root system $C\_2$

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 ( E₁, Φ₁) and ( E₂, Φ₂) 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 $\backslash langle\; x,\; y\; \backslash rangle$ is preserved.

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

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Weyl group The group of isometry of  E generated by reflections through hyperplanes associated to the roots of Φ is called the Weyl group of Φ. As it Faithful action on the finite set Φ, the Weyl group is always finite. The reflection planes are the hyperplanes perpendicular to the roots, indicated for $A\_2$ 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).

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Rank one example There is only one root system of rank 1, consisting of two nonzero vectors $\backslash \{\backslash alpha,\; -\backslash alpha\backslash \}$. This root system is called $A\_1$.

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Rank two examples In rank 2 there are four possibilities, corresponding to $\backslash sigma\_\backslash alpha(\backslash beta)\; =\; \backslash beta\; +\; n\backslash alpha$, where $n\; =\; 0,\; 1,\; 2,\; 3$. Proposition 8.8 The figure at right shows these possibilities, but with some redundancies: $A\_1\backslash times\; A\_1$ is isomorphic to $D\_2$ and $B\_2$ is isomorphic to $C\_2$.

Note that a root system is not determined by the lattice that it generates: $A\_1\; \backslash times\; A\_1$ and $B\_2$ both generate a square lattice while $A\_2$ and $G\_2$ both generate a hexagonal lattice.

Whenever Φ is a root system in E, and S is a Linear 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.

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Root systems arising from semisimple Lie algebras If $\backslash mathfrak\{g\}$ is a complex semisimple Lie algebra and $\backslash mathfrak\{h\}$ is a Cartan subalgebra, we can construct a root system as follows. We say that $\backslash alpha\backslash in\backslash mathfrak\{h\}^*$ is a root of $\backslash mathfrak\{g\}$ relative to $\backslash mathfrak\{h\}$ if $\backslash alpha\backslash neq\; 0$ and there exists some $X\backslash neq\; 0\backslash in\backslash mathfrak\{g\}$ such that $$H,X=\backslash alpha(H)X$$ for all $H\backslash in\backslash mathfrak\{h\}$. One can show that there is an inner product for which the set of roots forms a root system. The root system of $\backslash mathfrak\{g\}$ is a fundamental tool for analyzing the structure of $\backslash mathfrak\{g\}$ and classifying its representations. (See the section below on Root systems and Lie theory.)

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History The concept of a root system was originally introduced by Wilhelm Killing around 1889 (in German, Wurzelsystem). He used them in his attempt to classify all simple Lie algebras over the field of . (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 F₄. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.)

Killing investigated the structure of a Lie algebra $L$ by considering what is now called a Cartan subalgebra $\backslash mathfrak\{h\}$. Then he studied the roots of the characteristic polynomial $\backslash det\; (\backslash operatorname\{ad\}\_L\; x\; -\; t)$, where $x\; \backslash in\; \backslash mathfrak\{h\}$. Here a root is considered as a function of $\backslash mathfrak\{h\}$, or indeed as an element of the dual vector space $\backslash mathfrak\{h\}^*$. This set of roots forms a root system inside $\backslash mathfrak\{h\}^*$, as defined above, where the inner product is the Killing form.

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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 $\backslash langle\; \backslash beta,\; \backslash alpha\; \backslash rangle$ and $\backslash langle\; \backslash alpha,\; \backslash beta\; \backslash rangle$ are both integers, by assumption, and

Since $2\backslash cos(\backslash theta)\; \backslash in\; -2,2$, the only possible values for $\backslash cos(\backslash theta)$ are $0,\; \backslash pm\; \backslash tfrac\{1\}\{2\},\; \backslash pm\backslash tfrac\{\backslash sqrt\{2\}\}\{2\},\; \backslash pm\backslash tfrac\{\backslash sqrt\{3\}\}\{2\}$ and $\backslash pm\backslash tfrac\{\backslash sqrt\{4\}\}\{2\}\; =\; \backslash pm\; 1$, 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 2 α or −2 α, 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 $\backslash sqrt\{2\}$ and an angle of 30° or 150° corresponds to a length ratio of $\backslash sqrt\{3\}$.

In summary, here are the only possibilities for each pair of roots. Proposition 8.6

• 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 $\backslash sqrt\; 2$.
• Angle of 30 or 150 degrees, with a length ratio of $\backslash sqrt\; 3$.

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Positive roots and simple roots Given a root system $\backslash Phi$ we can always choose (in many ways) a set of positive roots. This is a subset $\backslash Phi^+$ of $\backslash Phi$ such that

• For each root $\backslash alpha\backslash in\backslash Phi$ exactly one of the roots $\backslash alpha$, $-\backslash alpha$ is contained in $\backslash Phi^+$.
• For any two distinct $\backslash alpha,\; \backslash beta\backslash in\; \backslash Phi^+$ such that $\backslash alpha+\backslash beta$ is a root, $\backslash alpha+\backslash beta\backslash in\backslash Phi^+$.

If a set of positive roots $\backslash Phi^+$ is chosen, elements of $-\backslash Phi^+$ are called negative roots. A set of positive roots may be constructed by choosing a hyperplane $V$ not containing any root and setting $\backslash Phi^+$ to be all the roots lying on a fixed side of $V$. Furthermore, every set of positive roots arises in this way.

An element of $\backslash Phi^+$ is called a simple root (also fundamental root) if it cannot be written as the sum of two elements of $\backslash Phi^+$. (The set of simple roots is also referred to as a base for $\backslash Phi$.) The set $\backslash Delta$ of simple roots is a basis of $E$ with the following additional special properties:

• Every root $\backslash alpha\backslash in\backslash Phi$ is a linear combination of elements of $\backslash Delta$ with integer coefficients.
• For each $\backslash alpha\backslash in\backslash Phi$, the coefficients in the previous point are either all non-negative or all non-positive.

For each root system $\backslash Phi$ 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.

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Dual root system, coroots, and integral elements

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The dual root system If Φ is a root system in E, the coroot α^∨ of a root α is defined by $$\backslash alpha^\backslash vee=\; \{2\backslash over\; (\backslash alpha,\backslash alpha)\}\backslash ,\; \backslash alpha.$$

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, $$(s\backslash alpha)^\backslash vee=\; s(\backslash alpha^\backslash vee).$$

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 $A\_n$ and $D\_n$ along with the exceptional root systems $E\_6,E\_7,E\_8,F\_4,G\_2$ are all self-dual, meaning that the dual root system is isomorphic to the original root system. By contrast, the $B\_n$ and $C\_n$ root systems are dual to one another, but not isomorphic (except when $n=2$).

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Integral elements A vector $\backslash lambda$ in E is called integral if its inner product with each coroot is an integer: $$2\backslash frac\{(\backslash lambda,\backslash alpha)\}\{(\backslash alpha,\backslash alpha)\}\backslash in\backslash mathbb\; Z,\backslash quad\backslash alpha\backslash in\backslash Phi.$$ Since the set of $\backslash alpha^\backslash vee$ with $\backslash alpha\backslash in\backslash Delta$ forms a base for the dual root system, to verify that $\backslash lambda$ is integral, it suffices to check the above condition for $\backslash alpha\backslash in\backslash Delta$.

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.

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Classification of root systems by Dynkin diagrams A root system is irreducible if it cannot be partitioned into the union of two proper subsets $\backslash Phi=\backslash Phi\_1\backslash cup\backslash Phi\_2$, such that $(\backslash alpha,\backslash beta)=0$ for all $\backslash alpha\backslash in\backslash Phi\_1$ and $\backslash beta\backslash in\backslash Phi\_2$ .

Irreducible root systems bijection to certain graphs, the named after Eugene Dynkin. The classification of these graphs is a simple matter of combinatorics, and induces a classification of irreducible root systems.

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Constructing the Dynkin diagram 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.)

• No edge if the vectors are orthogonal,
• An undirected single edge if they make an angle of 120 degrees,
• A directed double edge if they make an angle of 135 degrees, and
• A directed triple edge if they make an angle of 150 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, $\backslash sqrt\; 2$, $\backslash sqrt\; 3$. In the case of the $G\_2$ root system for example, there are two simple roots at an angle of 150 degrees (with a length ratio of $\backslash sqrt\; 3$). 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.)

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Classifying root systems Although a given root system has more than one possible set of simple roots, the Weyl group acts transitively on such choices.This follows from 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. A root systems is irreducible if and only if its Dynkin diagram is connected. 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 $\backslash Phi$ is a root system, the Dynkin diagram for the dual root system $\backslash Phi^\backslash vee$ is obtained from the Dynkin diagram of $\backslash Phi$ by keeping all the same vertices and edges, but reversing the directions of all arrows. Thus, we can see from their Dynkin diagrams that $B\_n$ and $C\_n$ are dual to each other.

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Weyl chambers and the Weyl group If $\backslash Phi\backslash subset\; E$ is a root system, we may consider the hyperplane perpendicular to each root $\backslash alpha$. Recall that $\backslash sigma\_\backslash alpha$ denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of $E$ generated by all the $\backslash sigma\_\backslash alpha$'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 $v\backslash in\; E$ such that $(\backslash alpha,v)>0$ for all $\backslash alpha\backslash in\backslash Delta$.

Since the reflections $\backslash sigma\_\backslash alpha,\backslash ,\backslash alpha\backslash in\backslash Phi$ preserve $\backslash Phi$, 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 $A\_2$ 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:

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.

In the $A\_2$ case, for example, the Weyl group has six elements and there are six Weyl chambers.

A related result is this one:

Theorem: Fix a Weyl chamber $C$. Then for all $v\backslash in\; E$, the Weyl-orbit of $v$ contains exactly one point in the closure $\backslash bar\; C$ of $C$.

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Root systems and Lie theory Irreducible root systems classify a number of related objects in Lie theory, notably the following:

• simple complex Lie algebras (see the discussion above on root systems arising from semisimple Lie algebras),
• simply connected complex Lie groups which are simple modulo centers, and
• simply connected compact group which are simple modulo centers.

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 $\backslash mathbb\; C$, following the arguments in Humphreys.See various parts of Chapters III, IV, and V of , culminating in Section 19 in Chapter V A preliminary result says that a semisimple Lie algebra is simple if and only if the associated root system is irreducible., Theorem 7.35 We thus restrict attention to irreducible root systems and simple Lie algebras.

• First, we must establish that for each simple algebra $\backslash mathfrak\; g$ there is only one root system. This assertion follows from the result that the Cartan subalgebra of $\backslash mathfrak\; g$ is unique up to automorphism, from which it follows that any two Cartan subalgebras give isomorphic root systems.
• Next, we need to show that for each irreducible root system, there can be at most one Lie algebra, that is, that the root system determines the Lie algebra up to isomorphism.
• Finally, we must show that for each irreducible root system, there is an associated simple Lie algebra. This claim is obvious for the root systems of type A, B, C, and D, for which the associated Lie algebras are the classical Lie algebras. It is then possible to analyze the exceptional algebras in a case-by-case fashion. Alternatively, one can develop a systematic procedure for building a Lie algebra from a root system, using . Section 18.3 and Theorem 18.4

For connections between the exceptional root systems and their Lie groups and Lie algebras see E₈, E₇, E₆, F₄, and G₂.

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Properties of the irreducible root systems

2 n n!
2 n n!
2 n−1 n!
51840
2903040
696729600
1152
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 , 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 r²/2 times the coroot lattice, where r is the length of a long root.

In the adjacent table, denotes the number of short roots, 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.

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Explicit construction of the irreducible root systems

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A_n

+ Simple roots in A3

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 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 n² + n roots in all. One choice of simple roots expressed in the standard basis is for .

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 A₂ root lattice is the vertex arrangement of the triangular tiling.

The A₃ root lattice is known to crystallographers as the face-centered cubic (or cubic close packed) lattice.

It is the vertex arrangement of the tetrahedral-octahedral honeycomb.

The A₃ root system (as well as the other rank-three root systems) may be modeled in the Zome. Section 8.9

In general, the A _n root lattice is the vertex arrangement of the n-dimensional simplicial honeycomb.

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B_n

+ Simple roots in B4

Let E = R ⁿ, and let Φ consist of all integer vectors in E of length 1 or . The total number of roots is 2 n². One choice of simple roots is for (the above choice of simple roots for A _n−1), and the shorter root .

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 + 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.

B₁ is isomorphic to A₁ via scaling by , and is therefore not a distinct root system.

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C_n

+ Simple roots in C4

Let E = R ⁿ, and let Φ consist of all integer vectors in E of length 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 n². 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.

C₂ is isomorphic to B₂ via scaling by and a 45 degree rotation, and is therefore not a distinct root system.

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D_n

+ Simple roots in D4

Let , and let Φ consist of all integer vectors in E of length . The total number of roots is . One choice of simple roots is for (the above choice of simple roots for ) together with .

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 vertices exist in the middle of the edges of the n-orthoplex.

D₃ coincides with A₃, and is therefore not a distinct root system. The twelve D₃ root vectors are expressed as the vertices of , a lower symmetry construction of the cuboctahedron.

D₄ has additional symmetry called triality. The twenty-four D₄ root vectors are expressed as the vertices of , a lower symmetry construction of the 24-cell.

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E₆, E₇, E₈

72 vertices of 122 represent the root vectors of E6 (Green nodes are doubled in this E6 Coxeter plane projection)	126 vertices of 231 represent the root vectors of E7	240 vertices of 421 represent the root vectors of E8

• The E₈ root system is any set of vectors in R⁸ that is congruent to the following set:$$$$

D_8 \cup \left\{ \frac 1 2 \left( \sum_{i=1}^8 \varepsilon_i \mathbf e_i \right) : \varepsilon_i = \pm1, \, \varepsilon_1 \cdots \varepsilon_8 = +1 \right\}.

The root system has 240 roots. The set just listed is the set of vectors of length in the E8 root lattice, also known simply as the E8 lattice or Γ₈. This is the set of points in R⁸ such that:

1. all the coordinates are or all the coordinates are (a mixture of integers and half-integers is not allowed), and
2. the sum of the eight coordinates is an even integer.

Thus, $$E\_8\; =\; \backslash left\backslash \{\; \backslash alpha\backslash in\backslash mathbb\; Z^8\; \backslash cup\; \backslash left(\backslash mathbb\; Z\; +\; \backslash tfrac\; 1\; 2\backslash right)^8\; :\; |\backslash alpha|^2\; =\; \backslash sum\backslash alpha\_i^2\; =\; 2,\backslash ,\; \backslash sum\backslash alpha\_i\; \backslash in\; 2\backslash mathbb\; Z.\; \backslash right\backslash \}$$

• The root system E₇ is the set of vectors in E₈ that are perpendicular to a fixed root in E₈. The root system E₇ has 126 roots.
• The root system E₆ is not the set of vectors in E₇ that are perpendicular to a fixed root in E₇, indeed, one obtains D₆ that way. However, E₆ is the subsystem of E₈ perpendicular to two suitably chosen roots of E₈. The root system E₆ has 72 roots.

+ Simple roots in E8: even coordinates
0
0
0
0
0
0
0
−

An alternative description of the E₈ lattice which is sometimes convenient is as the set Γ'₈ of all points in R⁸ 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 Γ₈ 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 Γ₈ is sometimes called the even coordinate system for E₈ while the lattice Γ'₈ is called the odd coordinate system.

One choice of simple roots for E₈ 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

α₇ = e₇ + e₆

(the above choice of simple roots for D₇) along with $$\backslash boldsymbol\backslash alpha\_8\; =\; \backslash boldsymbol\backslash beta\_0\; =\; -\backslash frac\{1\}\{2\}\; \backslash sum\_\{i=1\}^8\backslash mathbf\{e\}\_i\; =\; (-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2).$$

+ Simple roots in E8: odd coordinates
0
0
0
0
0
0
−1

One choice of simple roots for E₈ 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 A₇) along with

α₈ = β₅, where

$\backslash boldsymbol\backslash beta\_j\; =\; \backslash frac\{1\}\{2\}\; \backslash left(-\; \backslash sum\_\{i=1\}^j\; e\_i\; +\; \backslash sum\_\{i=j+1\}^8\; e\_i\backslash right).$

(Using β₃ would give an isomorphic result. Using β_1,7 or β_2,6 would simply give A₈ or D₈. As for β₄, 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 β₄ has coordinates (1,2,3,4,3,2,1) in the basis ( α _i).)

Since perpendicularity to α₁ means that the first two coordinates are equal, E₇ is then the subset of E₈ where the first two coordinates are equal, and similarly E₆ is the subset of E₈ where the first three coordinates are equal. This facilitates explicit definitions of E₇ and E₆ as

Note that deleting α₁ and then α₂ gives sets of simple roots for E₇ and E₆. However, these sets of simple roots are in different E₇ and E₆ subspaces of E₈ than the ones written above, since they are not orthogonal to α₁ or α₂.

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F₄

+ Simple roots in F4

For F₄, let E = R⁴, and let Φ denote the set of vectors α of length 1 or 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 B₃, plus $\backslash boldsymbol\backslash alpha\_4\; =\; -\backslash frac\{1\}\{2\}\; \backslash sum\_\{i=1\}^4\; e\_i$.

The F₄ root lattice—that is, the lattice generated by the F₄ root system—is the set of points in R⁴ such that either all the coordinates are or all the coordinates are (a mixture of integers and half-integers is not allowed). This lattice is isomorphic to the lattice of Hurwitz quaternions.

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G₂

+ Simple roots in G2

The root system G₂ has 12 roots, which form the vertices of a hexagram. See the picture above.

One choice of simple roots is ( α₁, β = α₂ − α₁) where α _i = e _i − e _i+1 for i = 1, 2 is the above choice of simple roots for A₂.

The G₂ root lattice—that is, the lattice generated by the G₂ roots—is the same as the A₂ root lattice.

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The root poset The set of positive roots is naturally ordered by saying that $\backslash alpha\; \backslash leq\; \backslash beta$ if and only if $\backslash beta-\backslash alpha$ is a nonnegative linear combination of simple roots. This poset is Graded poset by $\backslash deg\backslash left(\backslash sum\_\{\backslash alpha\; \backslash in\; \backslash Delta\}\; \backslash lambda\_\backslash alpha\; \backslash alpha\backslash right)\; =\; \backslash sum\_\{\backslash alpha\; \backslash in\; \backslash Delta\}\backslash lambda\_\backslash alpha$, 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. The Hasse graph is a visualization of the ordering of the root poset.

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See also

• ADE classification
• Affine root system
• Coxeter–Dynkin diagram
• Coxeter group
• Coxeter matrix
• Dynkin diagram
• root datum
• Semisimple Lie algebra
• Weights in the representation theory of semisimple Lie algebras
• Root system of a semi-simple Lie algebra
• Weyl group

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Notes

• . The classic reference for root systems.
• (1998). 9783540647676, Springer. . ISBN 9783540647676
• James Humphreys (1972). 9780387900537, Springer. . ISBN 9780387900537
• (1992). 9780521436137, Cambridge University Press. ISBN 9780521436137
• (1990). 9780521466936, Cambridge University Press. . ISBN 9780521466936
• (1998). 9780817640217, Birkhäuser. ISBN 9780817640217

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Further reading

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External links

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