Inmathematics, aroot system is a configuration ofvectors in aEuclidean space satisfying certain geometrical properties. The concept is fundamental in the theory ofLie groups andLie algebras, especially the classification and representation theory ofsemisimple Lie algebras. Since Lie groups (and some analogues such asalgebraic 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, byDynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such assingularity theory). Finally, root systems are important for their own sake, as inspectral graph theory.^[1]

As a first example, consider the six vectors in 2-dimensionalEuclidean space,R², as shown in the image at the right; call themroots. These vectorsspan the whole space. If you consider the lineperpendicular to any root, sayβ, then the reflection ofR² in that line sends any other root, sayα, to another root. Moreover, the root to which it is sent equalsα +nβ, wheren 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 asA₂.

LetE be a finite-dimensionalEuclideanvector space, with the standardEuclidean inner product denoted by${\displaystyle (\cdot ,\cdot )}$. Aroot system${\displaystyle \mathrm{\Phi }}$ inE is a finite set of non-zero vectors (calledroots) that satisfy the following conditions:^[2][3]

1. The rootsspanE.
2. The only scalar multiples of a root${\displaystyle \alpha \in \mathrm{\Phi }}$ that belong to${\displaystyle \mathrm{\Phi }}$ are${\displaystyle \alpha }$ itself and${\displaystyle -\alpha }$.
3. For every root${\displaystyle \alpha \in \mathrm{\Phi }}$, the set${\displaystyle \mathrm{\Phi }}$ is closed underreflection through thehyperplane perpendicular to${\displaystyle \alpha }$.
4. (Integrality) If${\displaystyle \alpha }$ and${\displaystyle \beta }$ are roots in${\displaystyle \mathrm{\Phi }}$, then the projection of${\displaystyle \beta }$ onto the line through${\displaystyle \alpha }$ is aninteger or half-integer multiple of${\displaystyle \alpha }$.

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

3. For any two roots${\displaystyle \alpha ,\beta \in \mathrm{\Phi }}$, the set${\displaystyle \mathrm{\Phi }}$ contains the element${\displaystyle {\sigma }_{\alpha }(\beta ):=\beta -2\frac{(\alpha ,\beta )}{(\alpha ,\alpha )}\alpha .}$
4. For any two roots${\displaystyle \alpha ,\beta \in \mathrm{\Phi }}$, the number${\displaystyle ⟨\beta ,\alpha ⟩:=2\frac{(\alpha ,\beta )}{(\alpha ,\alpha )}}$ is aninteger.

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 acrystallographic root system.^[5] Other authors omit condition 2; then they call root systems satisfying condition 2reduced.^[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 thatβ and its reflectionσ_α(β) differ by an integer multiple of α. Note that the operator$${\displaystyle ⟨\cdot ,\cdot ⟩:\mathrm{\Phi }\times \mathrm{\Phi }\to \mathbb{Z}}$$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${\displaystyle A_1\times A_1}$	Root system${\displaystyle D_2}$
Root system${\displaystyle A_2}$	Root system${\displaystyle G_2}$
Root system${\displaystyle B_2}$	Root system${\displaystyle C_2}$

Therank of a root system Φ is the dimension ofE. 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 systemsA₂,B₂, andG₂ pictured to the right, is said to beirreducible.

Two root systems (E₁, Φ₁) and (E₂, Φ₂) are calledisomorphic if there is an invertible linear transformationE₁ → E₂ which sends Φ₁ to Φ₂ such that for each pair of roots, the number${\displaystyle ⟨x,y⟩}$ is preserved.^[7]

Theroot lattice of a root system Φ is theZ-submodule ofE generated by Φ. It is alattice in E.

Thegroup ofisometries of E generated by reflections through hyperplanes associated to the roots of Φ is called theWeyl group of Φ. As itacts faithfully on the finite set Φ, the Weyl group is always finite. The reflection planes are the hyperplanes perpendicular to the roots, indicated for${\displaystyle A_2}$ 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).

There is only one root system of rank 1, consisting of two nonzero vectors${\displaystyle \{\alpha ,-\alpha \}}$. This root system is called${\displaystyle A_1}$.

In rank 2 there are four possibilities, corresponding to${\displaystyle {\sigma }_{\alpha }(\beta )=\beta +n\alpha }$, where${\displaystyle n=0,1,2,3}$.^[8] The figure at right shows these possibilities, but with some redundancies:${\displaystyle A_1\times A_1}$ is isomorphic to${\displaystyle D_2}$ and${\displaystyle B_2}$ is isomorphic to${\displaystyle C_2}$.

Note that a root system is not determined by the lattice that it generates:${\displaystyle A_1\times A_1}$ and${\displaystyle B_2}$ both generate asquare lattice while${\displaystyle A_2}$ and${\displaystyle G_2}$ both generate ahexagonal lattice.

Whenever Φ is a root system inE, andS is asubspace ofE 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.

If${\displaystyle \mathfrak{g}}$ is a complexsemisimple Lie algebra and${\displaystyle \mathfrak{h}}$ is aCartan subalgebra, we can construct a root system as follows. We say that${\displaystyle \alpha \in {\mathfrak{h}}^{\ast }}$ is aroot of${\displaystyle \mathfrak{g}}$ relative to${\displaystyle \mathfrak{h}}$ if${\displaystyle \alpha \ne 0}$ and there exists some${\displaystyle X\ne 0\in \mathfrak{g}}$ such that$${\displaystyle [H,X]=\alpha (H)X}$$for all${\displaystyle H\in \mathfrak{h}}$. One can show^[9] that there is an inner product for which the set of roots forms a root system. The root system of${\displaystyle \mathfrak{g}}$ is a fundamental tool for analyzing the structure of${\displaystyle \mathfrak{g}}$ and classifying its representations. (See the section below on Root systems and Lie theory.)

The concept of a root system was originally introduced byWilhelm Killing around 1889 (in German,Wurzelsystem^[10]).^[11] He used them in his attempt to classify allsimple Lie algebras over thefield ofcomplex 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 F₄. Cartan later corrected this mistake, by showing Killing's two root systems were isomorphic.^[12])

Killing investigated the structure of a Lie algebra${\displaystyle L}$ by considering what is now called aCartan subalgebra${\displaystyle \mathfrak{h}}$. Then he studied the roots of thecharacteristic polynomial${\displaystyle det({\mathrm{ad}}_{L}x-t)}$, where${\displaystyle x\in \mathfrak{h}}$. Here aroot is considered as a function of${\displaystyle \mathfrak{h}}$, or indeed as an element of the dual vector space${\displaystyle {\mathfrak{h}}^{\ast }}$. This set of roots forms a root system inside${\displaystyle {\mathfrak{h}}^{\ast }}$, as defined above, where the inner product is theKilling 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${\displaystyle ⟨\beta ,\alpha ⟩}$ and${\displaystyle ⟨\alpha ,\beta ⟩}$ are both integers, by assumption, and

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

In summary, here are the only possibilities for each pair of roots.^[13]

• 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${\displaystyle \sqrt{2}}$.
• Angle of 30 or 150 degrees, with a length ratio of${\displaystyle \sqrt{3}}$.

Given a root system${\displaystyle \mathrm{\Phi }}$ we can always choose (in many ways) a set ofpositive roots. This is a subset${\displaystyle {\mathrm{\Phi }}^{+}}$ of${\displaystyle \mathrm{\Phi }}$ such that

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

If a set of positive roots${\displaystyle {\mathrm{\Phi }}^{+}}$ is chosen, elements of${\displaystyle -{\mathrm{\Phi }}^{+}}$ are callednegative roots. A set of positive roots may be constructed by choosing a hyperplane${\displaystyle V}$ not containing any root and setting${\displaystyle {\mathrm{\Phi }}^{+}}$ to be all the roots lying on a fixed side of${\displaystyle V}$. Furthermore, every set of positive roots arises in this way.^[14]

An element of${\displaystyle {\mathrm{\Phi }}^{+}}$ is called asimple root (alsofundamental root) if it cannot be written as the sum of two elements of${\displaystyle {\mathrm{\Phi }}^{+}}$. (The set of simple roots is also referred to as abase for${\displaystyle \mathrm{\Phi }}$.) The set${\displaystyle \mathrm{\Delta }}$ of simple roots is a basis of${\displaystyle E}$ with the following additional special properties:^[15]

• Every root${\displaystyle \alpha \in \mathrm{\Phi }}$ is alinear combination of elements of${\displaystyle \mathrm{\Delta }}$ withinteger coefficients.
• For each${\displaystyle \alpha \in \mathrm{\Phi }}$, the coefficients in the previous point are either all non-negative or all non-positive.

For each root system${\displaystyle \mathrm{\Phi }}$ 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.^[16]

If Φ is a root system inE, thecoroot α^∨ of a root α is defined by$${\displaystyle {\alpha }^{\vee }=\frac{2}{(\alpha ,\alpha )}\alpha .}$$

The set of coroots also forms a root system Φ^∨ inE, called thedual root system (or sometimesinverse root system).By definition, α^{∨ ∨} = α, so that Φ is the dual root system of Φ^∨. The lattice inE spanned by Φ^∨ is called thecoroot lattice. Both Φ and Φ^∨ have the same Weyl groupW and, fors inW,$${\displaystyle (s\alpha {)}^{\vee }=s({\alpha }^{\vee }).}$$

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${\displaystyle A_n}$ and${\displaystyle D_n}$ along with the exceptional root systems${\displaystyle E_6,E_7,E_8,F_4,G_2}$ are all self-dual, meaning that the dual root system is isomorphic to the original root system. By contrast, the${\displaystyle B_n}$ and${\displaystyle C_n}$ root systems are dual to one another, but not isomorphic (except when${\displaystyle n=2}$).

A vector${\displaystyle \lambda }$ inE is calledintegral^[18] if its inner product with each coroot is an integer:$${\displaystyle 2\frac{(\lambda ,\alpha )}{(\alpha ,\alpha )}\in \mathbb{Z},\alpha \in \mathrm{\Phi }.}$$Since the set of${\displaystyle {\alpha }^{\vee }}$ with${\displaystyle \alpha \in \mathrm{\Delta }}$ forms a base for the dual root system, to verify that${\displaystyle \lambda }$ is integral, it suffices to check the above condition for${\displaystyle \alpha \in \mathrm{\Delta }}$.

The set of integral elements is called theweight lattice associated to the given root system. This term comes from therepresentation 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.

A root system is irreducible if it cannot be partitioned into the union of two proper subsets${\displaystyle \mathrm{\Phi }={\mathrm{\Phi }}_1\cup {\mathrm{\Phi }}_2}$, such that${\displaystyle (\alpha ,\beta )=0}$ for all${\displaystyle \alpha \in {\mathrm{\Phi }}_1}$ and${\displaystyle \beta \in {\mathrm{\Phi }}_2}$ .

Irreducible root systemscorrespond to certaingraphs, theDynkin diagrams named afterEugene Dynkin. The classification of these graphs is a simple matter ofcombinatorics, and induces a classification of irreducible root systems.

Given a root system, select a set Δ ofsimple 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,${\displaystyle \sqrt{2}}$,${\displaystyle \sqrt{3}}$. In the case of the${\displaystyle G_2}$ root system for example, there are two simple roots at an angle of 150 degrees (with a length ratio of${\displaystyle \sqrt{3}}$). 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.)

Although a given root system has more than one possible set of simple roots, theWeyl 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 diagram 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${\displaystyle \mathrm{\Phi }}$ is a root system, the Dynkin diagram for the dual root system${\displaystyle {\mathrm{\Phi }}^{\vee }}$ is obtained from the Dynkin diagram of${\displaystyle \mathrm{\Phi }}$ by keeping all the same vertices and edges, but reversing the directions of all arrows. Thus, we can see from their Dynkin diagrams that${\displaystyle B_n}$ and${\displaystyle C_n}$ are dual to each other.

If${\displaystyle \mathrm{\Phi }\subset E}$ is a root system, we may consider the hyperplane perpendicular to each root${\displaystyle \alpha }$. Recall that${\displaystyle {\sigma }_{\alpha }}$ denotes the reflection about the hyperplane and that theWeyl group is the group of transformations of${\displaystyle E}$ generated by all the${\displaystyle {\sigma }_{\alpha }}$'s. The complement of the set of hyperplanes is disconnected, and each connected component is called aWeyl chamber. If we have fixed a particular set Δ of simple roots, we may define thefundamental Weyl chamber associated to Δ as the set of points${\displaystyle v\in E}$ such that${\displaystyle (\alpha ,v)>0}$ for all${\displaystyle \alpha \in \mathrm{\Delta }}$.

Since the reflections${\displaystyle {\sigma }_{\alpha },\alpha \in \mathrm{\Phi }}$ preserve${\displaystyle \mathrm{\Phi }}$, 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${\displaystyle A_2}$ 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:^[22]

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${\displaystyle A_2}$ case, for example, the Weyl group has six elements and there are six Weyl chambers.

A related result is this one:^[23]

Theorem: Fix a Weyl chamber${\displaystyle C}$. Then for all${\displaystyle v\in E}$, the Weyl-orbit of${\displaystyle v}$ contains exactly one point in the closure${\displaystyle \overline{C}}$ of${\displaystyle C}$.

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 connectedcompact Lie groups which are simple modulo centers.

In each case, the roots are non-zeroweights of theadjoint representation.

We now give a brief indication of how irreducible root systems classify simple Lie algebras over${\displaystyle \mathbb{C}}$, following the arguments in Humphreys.^[24] A preliminary result says that asemisimple 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.

• First, we must establish that for each simple algebra${\displaystyle \mathfrak{g}}$ there is only one root system. This assertion follows from the result that the Cartan subalgebra of${\displaystyle \mathfrak{g}}$ is unique up to automorphism,^[26] 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.^[27]
• 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 theclassical 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, usingSerre's relations.^[28]

For connections between the exceptional root systems and their Lie groups and Lie algebras seeE₈,E₇,E₆,F₄, andG₂.

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 theclassical root systems) and five exceptional cases (theexceptional 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 toshort andlong roots. If all roots have the same length they are taken to be long by definition and the root system is said to besimply 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 tor²/2 times the coroot lattice, wherer 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 theCartan matrix, and |W| denotes the order of theWeyl group.

Simple roots inA₃

	e1	e2	e3	e4
α1	1	−1	0	0
α2	0	1	−1	0
α3	0	0	1	−1

LetE be the subspace ofRⁿ⁺¹ for which the coordinates sum to 0, and let Φ be the set of vectors inE of length√2 and which areinteger vectors, i.e. have integer coordinates inRⁿ⁺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 aren² +n roots in all. One choice of simple roots expressed in thestandard basis isα_i =e_i −e_i+1 for1 ≤i ≤n.

Thereflectionσ_i through thehyperplane perpendicular toα_i is the same aspermutation of the adjacentith and (i + 1)thcoordinates. Suchtranspositions generate the fullpermutation 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; butreflection of a simple root perpendicular to a nonadjacent simple root leaves it unchanged, differing by a multiple of 0.

TheA_n root lattice – that is, the lattice generated by theA_n roots – is most easily described as the set of integer vectors inRⁿ⁺¹ whose components sum to zero.

TheA₂ root lattice is thevertex arrangement of thetriangular tiling.

TheA₃ root lattice is known to crystallographers as theface-centered cubic (orcubic close packed) lattice.^[29] It is the vertex arrangement of thetetrahedral-octahedral honeycomb.

TheA₃ root system (as well as the other rank-three root systems) may be modeled in theZometool construction set.^[30]

In general, theA_n root lattice is the vertex arrangement of then-dimensionalsimplicial honeycomb.

Simple roots inB₄

	e1	e2	e3	e4
α1	1	−1	0	0
α2	0	1	−1	0
α3	0	0	1	−1
α4	0	0	0	1

LetE =Rⁿ, and let Φ consist of all integer vectors inE of length 1 or√2. The total number of roots is 2n². One choice of simple roots isα_i =e_i –e_i+1 for1 ≤i ≤n – 1 (the above choice of simple roots forA_n−1), and the shorter rootα_n =e_n.

The reflectionσ_n through the hyperplane perpendicular to the short rootα_n is of course simply negation of thenth coordinate. For the long simple rootα_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.

TheB_n root lattice—that is, the lattice generated by theB_n roots—consists of all integer vectors.

B₁ is isomorphic toA₁ via scaling by√2, and is therefore not a distinct root system.

Simple roots inC₄

	e1	e2	e3	e4
α1	1	−1	0	0
α2	0	1	−1	0
α3	0	0	1	−1
α4	0	0	0	2

LetE =Rⁿ, and let Φ consist of all integer vectors inE of length√2 together with all vectors of the form 2λ, whereλ is an integer vector of length 1. The total number of roots is 2n². One choice of simple roots is:α_i =e_i −e_i+1, for 1 ≤i ≤n − 1 (the above choice of simple roots forA_n−1), and the longer rootα_n = 2e_n.The reflectionσ_n(α_n−1) =α_n−1 +α_n, butσ_n−1(α_n) =α_n + 2α_n−1.

TheC_n root lattice—that is, the lattice generated by theC_n roots—consists of all integer vectors whose components sum to an even integer.

C₂ is isomorphic toB₂ via scaling by√2 and a 45 degree rotation, and is therefore not a distinct root system.

Simple roots inD₄

	e1	e2	e3	e4
α1	1	−1	0	0
α2	0	1	−1	0
α3	0	0	1	−1
α4	0	0	1	1

LetE =Rⁿ, and let Φ consist of all integer vectors inE of length√2. The total number of roots is2n(n − 1). One choice of simple roots isα_i =e_i −e_i+1 for1 ≤i ≤n − 1 (the above choice of simple roots forA_n−1) together withα_n =e_n−1 +e_n.

Reflection through the hyperplane perpendicular toα_n is the same astransposing and negating the adjacentn-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.

TheD_n root lattice – that is, the lattice generated by theD_n roots – consists of all integer vectors whose components sum to an even integer. This is the same as theC_n root lattice.

TheD_n roots are expressed as the vertices of a rectifiedn-orthoplex,Coxeter–Dynkin diagram:.... The2n(n − 1) vertices exist in the middle of the edges of then-orthoplex.

D₃ coincides withA₃, and is therefore not a distinct root system. The twelveD₃ root vectors are expressed as the vertices of, a lower symmetry construction of thecuboctahedron.

D₄ has additional symmetry calledtriality. The twenty-fourD₄ root vectors are expressed as the vertices of, a lower symmetry construction of the24-cell.

72 vertices of122 represent the root vectors ofE6 (Green nodes are doubled in this E6 Coxeter plane projection)	126 vertices of231 represent the root vectors ofE7	240 vertices of421 represent the root vectors ofE8

• TheE₈ root system is any set of vectors inR⁸ that iscongruent to the following set:$${\displaystyle D_8\cup \left\{\frac{1}{2}\left(\sum _{i=1}^8{\epsilon }_i{\mathbf{e}}_i\right):{\epsilon }_i=\pm 1,{\epsilon }_1\cdots {\epsilon }_8=+1\right\}.}$$

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 theE8 lattice or Γ₈. This is the set of points inR⁸ such that:

1. all the coordinates areintegers or all the coordinates arehalf-integers (a mixture of integers and half-integers is not allowed), and
2. the sum of the eight coordinates is aneven integer.

Thus,$${\displaystyle E_8=\left\{\alpha \in {\mathbb{Z}}^8\cup {\left(\mathbb{Z}+\frac{1}{2}\right)}^8:|\alpha {|}^2=\sum {\alpha }_i^2=2,\sum {\alpha }_i\in 2\mathbb{Z}.\right\}}$$

• The root systemE₇ is the set of vectors inE₈ that are perpendicular to a fixed root inE₈. The root systemE₇ has 126 roots.
• The root systemE₆ is not the set of vectors inE₇ that are perpendicular to a fixed root inE₇, indeed, one obtainsD₆ that way. However,E₆ is the subsystem ofE₈ perpendicular to two suitably chosen roots ofE₈. The root systemE₆ has 72 roots.

An alternative description of theE₈ lattice which is sometimes convenient is as the set Γ'₈ of all points inR⁸ 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 areisomorphic; one may pass from one to the other by changing the signs of any odd number of coordinates. The lattice Γ₈ is sometimes called theeven coordinate system forE₈ while the lattice Γ'₈ is called theodd coordinate system.

One choice of simple roots forE₈ 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 forD₇) along with$${\displaystyle {\mathit{\alpha }}_8={\mathit{\beta }}_0=-\frac{1}{2}\sum _{i=1}^8{\mathbf{e}}_i=(-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2).}$$

One choice of simple roots forE₈ 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 forA₇) along with

α₈ =β₅, where

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

(Usingβ₃ would give an isomorphic result. Usingβ_1,7 orβ_2,6 would simply giveA₈ orD₈. As forβ₄, its coordinates sum to 0, and the same is true forα_1...7, so they span only the 7-dimensional subspace for which the coordinates sum to 0; in fact −2β₄ has coordinates (1,2,3,4,3,2,1) in the basis (α_i).)

Since perpendicularity toα₁ means that the first two coordinates are equal,E₇ is then the subset ofE₈ where the first two coordinates are equal, and similarlyE₆ is the subset ofE₈ where the first three coordinates are equal. This facilitates explicit definitions ofE₇ andE₆ as

E₇ = {α ∈Z⁷ ∪ (Z+1/2)⁷ : Σα_i² +α₁² = 2, Σα_i +α₁ ∈ 2Z},

E₆ = {α ∈Z⁶ ∪ (Z+1/2)⁶ : Σα_i² + 2α₁² = 2, Σα_i + 2α₁ ∈ 2Z}

Note that deletingα₁ and thenα₂ gives sets of simple roots forE₇ andE₆. However, these sets of simple roots are in differentE₇ andE₆ subspaces ofE₈ than the ones written above, since they are not orthogonal toα₁ orα₂.

Simple roots inF₄

	e1	e2	e3	e4
α1	1	−1	0	0
α2	0	1	−1	0
α3	0	0	1	0
α4	−⁠1/2⁠	−⁠1/2⁠	−⁠1/2⁠	−⁠1/2⁠

ForF₄, letE =R⁴, 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 forB₃, plus${\mathit{\alpha }}_4=-\frac{1}{2}\sum _{i=1}^4{e}_i$.

TheF₄ root lattice—that is, the lattice generated by theF₄ root system—is the set of points inR⁴ such that either all the coordinates areintegers or all the coordinates arehalf-integers (a mixture of integers and half-integers is not allowed). This lattice is isomorphic to the lattice ofHurwitz quaternions.

Simple roots inG₂

	e1	e2	e3
α1	1	−1	0
β	−1	2	−1

The root systemG₂ has 12 roots, which form the vertices of ahexagram. See the pictureabove.

One choice of simple roots is (α₁,β =α₂ −α₁) whereα_i =e_i −e_i+1 fori = 1, 2 is the above choice of simple roots forA₂.

TheG₂ root lattice—that is, the lattice generated by theG₂ roots—is the same as theA₂ root lattice.

The set of positive roots is naturally ordered by saying that${\displaystyle \alpha \le \beta }$ if and only if${\displaystyle \beta -\alpha }$ is a nonnegative linear combination of simple roots. Thisposet isgraded by$\mathrm{deg}\left(\sum _{\alpha \in \mathrm{\Delta }}{\lambda }_{\alpha }\alpha \right)=\sum _{\alpha \in \mathrm{\Delta }}{\lambda }_{\alpha }$, 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.^[31] The Hasse graph is a visualization of the ordering of the root poset.

• 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

• Adams, J.F. (1983),Lectures on Lie groups, University of Chicago Press,ISBN 0-226-00530-5
• Bourbaki, Nicolas (2002),Lie groups and Lie algebras, Chapters 4–6 (translated from the 1968 French original by Andrew Pressley), Elements of Mathematics, Springer-Verlag,ISBN 3-540-42650-7. The classic reference for root systems.
• Bourbaki, Nicolas (1998).Elements of the History of Mathematics. Springer.ISBN 3540647678.
• Coleman, A.J. (Summer 1989), "The greatest mathematical paper of all time",The Mathematical Intelligencer,11 (3):29–38,doi:10.1007/bf03025189,S2CID 35487310
• Hall, Brian C. (2015),Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,ISBN 978-3319134666
• Humphreys, James (1972).Introduction to Lie algebras and Representation Theory. Springer.ISBN 0387900535.
• Humphreys, James (1992).Reflection Groups and Coxeter Groups. Cambridge University Press.ISBN 0521436133.
• Killing, Wilhelm (June 1888)."Die Zusammensetzung der stetigen endlichen Transformationsgruppen".Mathematische Annalen.31 (2):252–290.doi:10.1007/BF01211904.S2CID 120501356. Archived fromthe original on 2016-03-05.
  • — (March 1888)."Part 2".Math. Ann.33 (1):1–48.doi:10.1007/BF01444109.S2CID 124198118.
  • — (March 1889)."Part 3".Math. Ann.34 (1):57–122.doi:10.1007/BF01446792.S2CID 179177899. Archived fromthe original on 2015-02-21.
  • — (June 1890)."Part 4".Math. Ann.36 (2):161–189.doi:10.1007/BF01207837.S2CID 179178061.
• Kac, Victor G. (1990).Infinite-Dimensional Lie Algebras (3rd ed.). Cambridge University Press.ISBN 978-0-521-46693-6.
• Springer, T.A. (1998).Linear Algebraic Groups (2nd ed.). Birkhäuser.ISBN 0817640215.

• Dynkin, E.B. (1947)."The structure of semi-simple algebras".Uspekhi Mat. Nauk. 2 (in Russian).4 (20):59–127.MR 0027752.

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