Root Systems and Dynkin Diagrams

⚡ Key Takeaways

Root systems are finite configurations of vectors in a Euclidean space that are stable under the reflections they generate; Dynkin diagrams encode them combinatorially as decorated graphs. Key result: irreducible reduced root systems are classified exactly by the connected Dynkin diagrams of types An,Bn,Cn,Dn,E6,E7,E8,F4,G2A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2. Applies to: semisimple Lie algebras and groups, Weyl groups, Coxeter groups, singularity theory, representation theory, and parts of combinatorics and physics. Originated: Wilhelm Killing (1888–1890) and Élie Cartan (1894), with the diagrammatic encoding due to Eugene Dynkin (1947).

💡 The Core Insight

All the structure lives in the Cartan integers ⟨β,α⟩∈Z\langle\beta,\alpha\rangle\in\mathbb{Z}. Because ⟨α,β⟩⟨β,α⟩=4cos⁡2θ∈{0,1,2,3}\langle\alpha,\beta\rangle\langle\beta,\alpha\rangle = 4\cos^2\theta \in \{0,1,2,3\} for non-proportional roots, the angle and length-ratio data of any pair of roots is quantized. A base (simple system) selects a minimal spanning subset such that every root is a Z≥0\mathbb{Z}_{\geq0}- or Z≤0\mathbb{Z}_{\leq0}-combination of it; the pairwise Cartan data of the base, drawn as a decorated graph, is the Dynkin diagram. The classification is then pure graph combinatorics: which connected diagrams can occur.

Formal Statement

DefinitionRoot System

Let EE be a finite-dimensional real vector space with positive-definite inner product (⋅,⋅)(\cdot,\cdot). For a nonzero α∈E\alpha\in E define the reflection

1
sα(v)=v−2(v,α)(α,α) α=v−⟨v,α⟩ α,\quad s_\alpha(v) = v - \frac{2(v,\alpha)}{(\alpha,\alpha)}\,\alpha = v - \langle v,\alpha\rangle\,\alpha,

Derivation hint: sαs_\alpha fixes the hyperplane α⊥\alpha^\perp and sends α↦−α\alpha\mapsto-\alpha; subtract twice the projection of vv onto α\alpha. where the Cartan pairing (linear in the first slot only) is

2
⟨v,α⟩:=2(v,α)(α,α).\quad \langle v,\alpha\rangle := \frac{2(v,\alpha)}{(\alpha,\alpha)}.

Derivation hint: this is 2×2\times the component of vv along the unit vector α/∥α∥\alpha/\|\alpha\| divided by ∥α∥\|\alpha\|, normalized so ⟨α,α⟩=2\langle\alpha,\alpha\rangle=2. A finite set Φ⊂E∖{0}\Phi\subset E\setminus\{0\} is a root system if:

  • (R1) Φ\Phi spans EE;
  • (R2) for α∈Φ\alpha\in\Phi,   Rα∩Φ={±α}\;\mathbb{R}\alpha\cap\Phi=\{\pm\alpha\} (reduced condition);
  • (R3) sα(Φ)=Φs_\alpha(\Phi)=\Phi for all α∈Φ\alpha\in\Phi;
  • (R4) ⟨β,α⟩∈Z\langle\beta,\alpha\rangle\in\mathbb{Z} for all α,β∈Φ\alpha,\beta\in\Phi.

The elements of Φ\Phi are roots; dim⁡E\dim E is the rank.

DefinitionCoroots, Base, Weyl Group

For α∈Φ\alpha\in\Phi, the coroot is α∨=2α/(α,α)\alpha^\vee = 2\alpha/(\alpha,\alpha), so ⟨v,α⟩=(v,α∨)\langle v,\alpha\rangle = (v,\alpha^\vee) and {α∨}=Φ∨\{\alpha^\vee\}=\Phi^\vee forms the dual root system. The Weyl group is the finite group

3
W(Φ)=⟨ sα:α∈Φ ⟩  ≤  O(E).\quad W(\Phi) = \langle\, s_\alpha : \alpha\in\Phi \,\rangle \;\leq\; O(E).

Derivation hint: each sαs_\alpha is an orthogonal involution permuting the finite spanning set Φ\Phi, so WW injects into the symmetric group on Φ\Phi and is finite. A base (simple system) Δ={α1,…,αℓ}⊂Φ\Delta=\{\alpha_1,\dots,\alpha_\ell\}\subset\Phi is a basis of EE such that every β∈Φ\beta\in\Phi writes as β=∑iciαi\beta=\sum_i c_i\alpha_i with all ci∈Z≥0c_i\in\mathbb{Z}_{\geq0} (a positive root) or all ci∈Z≤0c_i\in\mathbb{Z}_{\leq0} (a negative root). Bases exist and are permuted simply transitively by WW.

TheoremAngle and Length Quantization

For non-proportional roots α,β∈Φ\alpha,\beta\in\Phi with angle θ\theta,

4
⟨α,β⟩ ⟨β,α⟩=4cos⁡2θ∈{0,1,2,3}.\quad \langle\alpha,\beta\rangle\,\langle\beta,\alpha\rangle = 4\cos^2\theta \in \{0,1,2,3\}.

Derivation hint: multiply the two Cartan integers; the cross-products of inner products combine to 4(α,β)2/[(α,α)(β,β)]=4cos⁡2θ4(\alpha,\beta)^2/[(\alpha,\alpha)(\beta,\beta)] = 4\cos^2\theta. Both factors are integers of the same sign, and cos⁡2θ<1\cos^2\theta<1 excludes 44. Consequently the only possible angles are θ∈{90∘,120∘,135∘,150∘}\theta\in\{90^\circ,120^\circ,135^\circ,150^\circ\} (plus 60∘,45∘,30∘60^\circ,45^\circ,30^\circ for the supplement), and when θ≠90∘\theta\neq90^\circ the squared length ratio (β,β)/(α,α)(\beta,\beta)/(\alpha,\alpha) is forced to 1,2,1,2, or 33.

TheoremCartan Matrix

Fix a base Δ={α1,…,αℓ}\Delta=\{\alpha_1,\dots,\alpha_\ell\}. The Cartan matrix is

5
Aij=⟨αi,αj⟩=2(αi,αj)(αj,αj).\quad A_{ij} = \langle\alpha_i,\alpha_j\rangle = \frac{2(\alpha_i,\alpha_j)}{(\alpha_j,\alpha_j)}.

Derivation hint: apply the Cartan pairing [2] to ordered pairs of simple roots; diagonal entries are ⟨αi,αi⟩=2\langle\alpha_i,\alpha_i\rangle=2 by definition. It satisfies: Aii=2A_{ii}=2; Aij∈{0,−1,−2,−3}A_{ij}\in\{0,-1,-2,-3\} for i≠ji\neq j; Aij=0  ⟺  Aji=0A_{ij}=0\iff A_{ji}=0; and AA is symmetrizable to a positive-definite matrix. The root system is recovered (up to isomorphism) from AA, hence from the Dynkin diagram.

DefinitionDynkin Diagram

The Dynkin diagram of (Φ,Δ)(\Phi,\Delta) is the labeled graph with one node per simple root αi\alpha_i, and between αi,αj\alpha_i,\alpha_j a number of edges equal to

6
nij=⟨αi,αj⟩⟨αj,αi⟩=AijAji∈{0,1,2,3}.\quad n_{ij} = \langle\alpha_i,\alpha_j\rangle\langle\alpha_j,\alpha_i\rangle = A_{ij}A_{ji}\in\{0,1,2,3\}.

Derivation hint: this is the integer 4cos⁡2θ4\cos^2\theta from [4]; it counts the edge multiplicity (0 = no edge, 1 = single, 2 = double, 3 = triple). When nij>1n_{ij}>1 (the two roots have different lengths) an arrow points from the long root toward the short root. The diagram is connected iff Φ\Phi is irreducible (not an orthogonal direct sum).

TheoremClassification — Killing–Cartan

Every connected Dynkin diagram of a reduced irreducible root system is isomorphic to exactly one of the following:

7
Aℓ (ℓ≥1),  Bℓ (ℓ≥2),  Cℓ (ℓ≥3),  Dℓ (ℓ≥4),  E6,E7,E8,F4,G2.\quad A_\ell\,(\ell\ge1),\;B_\ell\,(\ell\ge2),\;C_\ell\,(\ell\ge3),\;D_\ell\,(\ell\ge4),\;E_6,E_7,E_8,F_4,G_2.

Derivation hint: positive-definiteness of the (symmetrized) Cartan matrix restricts which subgraphs (cycles, high-valence nodes, long chains of multiple edges) can occur; eliminating the forbidden configurations leaves exactly this list. Each diagram arises from a unique irreducible root system, and the simply-laced ones (A,D,EA,D,E: all roots same length) have WW generated by reflections in equal-length roots.

Figure
The nine connected Dynkin diagrams of irreducible reduced root systems

```svg

··· A_n ··· B_n (long→short) ··· C_n ··· D_n E_8 (E_6,E_7 truncate) F_4 G_2 (triple edge)

```

Why It Holds / Derivation

ProofSketch of the Classification

Step 1 — Quantize angles. By [4], for non-proportional α,β\alpha,\beta the product ⟨α,β⟩⟨β,α⟩∈{0,1,2,3}\langle\alpha,\beta\rangle\langle\beta,\alpha\rangle\in\{0,1,2,3\}. Listing the integer solutions of ⟨α,β⟩⟨β,α⟩=4cos⁡2θ\langle\alpha,\beta\rangle\langle\beta,\alpha\rangle = 4\cos^2\theta with ∣⟨⋅⟩∣≤3|\langle\cdot\rangle|\le3 yields the finite menu of (angle, length-ratio) pairs. This is the only arithmetic input.

Step 2 — Reduce to a base. Choose a generic linear functional ff; declare a root positive if f(α)>0f(\alpha)>0. The indecomposable positive roots (not a sum of two positive roots) form a base Δ\Delta. One shows ∣Δ∣=dim⁡E|\Delta|=\dim E and every root is a nonneg/nonpos integer combination.

Step 3 — Simple roots make obtuse angles. For distinct αi,αj∈Δ\alpha_i,\alpha_j\in\Delta, (αi,αj)≤0(\alpha_i,\alpha_j)\le0, so θ≥90∘\theta\ge90^\circ and nij=4cos⁡2θ∈{0,1,2,3}n_{ij}=4\cos^2\theta\in\{0,1,2,3\}. Encode (Δ,A)(\Delta,A) as the Dynkin diagram.

Step 4 — Positive-definiteness constraints. The Gram matrix of Δ\Delta is positive-definite. Define for each αi\alpha_i the unit vector εi=αi/∥αi∥\varepsilon_i=\alpha_i/\|\alpha_i\|; then for any subset ∑ciεi\sum c_i\varepsilon_i one has ∥∑ciεi∥2>0\|\sum c_i\varepsilon_i\|^2>0. Working out this inequality forbids:

8
∑i<j2 cicj(εi,εj)>−∑ici2⟹bounded edge multiplicities and valences.\quad \sum_{i<j} 2\,c_ic_j(\varepsilon_i,\varepsilon_j) > -\sum_i c_i^2 \quad\Longrightarrow\quad \text{bounded edge multiplicities and valences.}

Derivation hint: expand ∥∑ciεi∥2=∑ci2+2∑i<jcicj(εi,εj)\|\sum c_i\varepsilon_i\|^2 = \sum c_i^2 + 2\sum_{i<j}c_ic_j(\varepsilon_i,\varepsilon_j) and demand positivity; with (εi,εj)=−12nij(\varepsilon_i,\varepsilon_j)=-\tfrac12\sqrt{n_{ij}} this gives concrete combinatorial bans.

Step 5 — Eliminate forbidden subgraphs. Positive-definiteness implies: (a) no cycles; (b) at most one node of valence 3 (a branch node) and no node of valence ≥4\ge4; (c) at most one multiple edge, which (if present) excludes any branch node; (d) bounds on the lengths of the three arms emanating from a branch node, via the "balance" inequality. The surviving connected diagrams are precisely the list [7].

Step 6 — Realizability. Each surviving diagram is realized by an explicit root system (e.g. AℓA_\ell by {ei−ej}⊂Rℓ+1\{e_i-e_j\}\subset\mathbb{R}^{\ell+1}), completing the bijection. □\square

∎
mathematician · Wilhelm Killing & Élie Cartan | 1847–1923 / 1869–1951

Killing, working in near-isolation in Braunsberg, sought to classify the "infinitesimal groups" (Lie algebras) and in 1888–1890 produced the first classification of simple Lie algebras over C\mathbb{C}, discovering the exceptional types and inventing what we now call roots, the Cartan matrix, and the characteristic equation. His papers were riddled with errors and obscure, and he doubted his own exceptional cases. Élie Cartan, in his 1894 thesis, corrected, completed, and rigorized the entire theory, gave clean existence proofs, and built the bridge to Lie groups and differential geometry. The phrase "Cartan integer" and "Cartan matrix" honor him; Hermann Weyl later supplied the global group-theoretic and representation-theoretic superstructure (the Weyl character formula, integrality of weights). Eugene Dynkin's 1947 graphical encoding turned the classification into a teachable combinatorial picture.

Building Intuition

🧠 Intuition

Think of a root system as a kaleidoscope. The reflecting hyperplanes α⊥\alpha^\perp chop EE into congruent chambers (Weyl chambers); the Weyl group shuffles them simply transitively. A base Δ\Delta picks one chamber, and the simple roots are the inward-pointing normals to its walls. The whole infinite-looking symmetry is generated by bouncing off just ℓ\ell mirrors. The integrality (R4) is the secret enforcer: it says roots "see each other at quantized angles." Two mirrors can only meet at 90∘,60∘,45∘,90^\circ, 60^\circ, 45^\circ, or 30∘30^\circ — exactly the dihedral angles that tile space. That is why the answer is a finite list: there are only so many ways to arrange a handful of mutually-quantized mirrors in a positive-definite space without the geometry "blowing up" (which positive-definiteness forbids). The exceptional groups G2,F4,E8G_2,F_4,E_8 are the rare maximal configurations where this packing is just barely possible.

Worked Examples

ExampleThe Root System A2A_2 (rank 2)

Given: E={x∈R3:∑xi=0}E=\{x\in\mathbb{R}^3:\sum x_i=0\}, roots Φ={ei−ej:i≠j}\Phi=\{e_i-e_j: i\neq j\}, six in total. Take base Δ={α1=e1−e2, α2=e2−e3}\Delta=\{\alpha_1=e_1-e_2,\ \alpha_2=e_2-e_3\}.

Steps: 1. Inner products: (α1,α1)=(α2,α2)=2(\alpha_1,\alpha_1)=(\alpha_2,\alpha_2)=2, and (α1,α2)=−1(\alpha_1,\alpha_2)=-1. 2. Cartan integers: ⟨α1,α2⟩=2(−1)/2=−1\langle\alpha_1,\alpha_2\rangle = 2(-1)/2 = -1; similarly ⟨α2,α1⟩=−1\langle\alpha_2,\alpha_1\rangle=-1. 3. Cartan matrix A=(2−1−12)A=\begin{pmatrix}2&-1\\-1&2\end{pmatrix}, edge multiplicity n12=(−1)(−1)=1n_{12}=(-1)(-1)=1 → single edge, no arrow (equal lengths). 4. Angle: 4cos⁡2θ=1⇒θ=120∘4\cos^2\theta=1\Rightarrow\theta=120^\circ. 5. Weyl group: ∣W∣=6|W|=6, the symmetric group S3S_3 acting on the 6 roots; the diagram is two nodes joined by one edge.

Answer: A2: ∘ ⁣− ⁣∘,  A=(2−1−12),  ∣Φ∣=6, ∣W∣=6\boxed{A_2:\ \circ\!-\!\circ,\ \ A=\begin{pmatrix}2&-1\\-1&2\end{pmatrix},\ \ |\Phi|=6,\ |W|=6} — the Lie algebra sl3(C)\mathfrak{sl}_3(\mathbb{C}).

ExampleThe Root System G2G_2 (rank 2, triple edge)

Given: a rank-2 system with a short root α\alpha ((α,α)=2(\alpha,\alpha)=2) and a long root β\beta ((β,β)=6(\beta,\beta)=6) at angle 150∘150^\circ. Base Δ={α,β}\Delta=\{\alpha,\beta\}.

Steps: 1. (α,β)=∥α∥∥β∥cos⁡150∘=26(−32)=−3(\alpha,\beta)=\|\alpha\|\|\beta\|\cos150^\circ = \sqrt2\sqrt6(-\tfrac{\sqrt3}{2}) = -3. 2. ⟨α,β⟩=2(−3)/6=−1\langle\alpha,\beta\rangle = 2(-3)/6 = -1; ⟨β,α⟩=2(−3)/2=−3\langle\beta,\alpha\rangle = 2(-3)/2 = -3. 3. Edge multiplicity n=(−1)(−3)=3n=(-1)(-3)=3 → triple edge, arrow from long β\beta to short α\alpha. 4. Generating reflections produce ∣Φ∣=12|\Phi|=12 roots (6 short, 6 long) and ∣W∣=12|W|=12 (dihedral of order 12). 5. Cartan matrix A=(2−1−32)A=\begin{pmatrix}2&-1\\-3&2\end{pmatrix}, det⁡A=4−3=1\det A = 4-3 = 1.

Answer: G2: ∘ ⁣⇛ ⁣∘,  A=(2−1−32),  ∣Φ∣=12, ∣W∣=12\boxed{G_2:\ \circ\!\Rrightarrow\!\circ,\ \ A=\begin{pmatrix}2&-1\\-3&2\end{pmatrix},\ \ |\Phi|=12,\ |W|=12} — the smallest exceptional Lie algebra, dimension 14.

Common Misconceptions

⚠️ Common mistake

The Cartan pairing ⟨v,α⟩\langle v,\alpha\rangle is NOT symmetric. It is linear in vv and "anti-scales" in α\alpha. For roots of different lengths, ⟨α,β⟩≠⟨β,α⟩\langle\alpha,\beta\rangle\neq\langle\beta,\alpha\rangle — that asymmetry is exactly what the arrow in a multiply-laced diagram records. Confusing the two flips Bn↔CnB_n\leftrightarrow C_n.

A Dynkin diagram is not just a Coxeter diagram. The undirected Coxeter diagram records only angles (nijn_{ij}), so BnB_n and CnC_n have the same Coxeter diagram and Weyl group. The arrow (length data) distinguishes them, and is essential to recover the root system and Lie algebra. BnB_n and CnC_n are non-isomorphic root systems that are dual: Bn∨=CnB_n^\vee=C_n.

A1A_1, B1=C1B_1=C_1, D1D_1, D2=A1×A1D_2=A_1\times A_1, D3=A3D_3=A_3, B2=C2B_2=C_2 "coincidences" are why the index ranges in [7] are restricted — they prevent double-counting the same diagram.

Counterexample

A finite reflection-symmetric vector set need not be a root system. Take the 8 vertices of a regular octagon (vectors at 45∘45^\circ increments) in R2\mathbb{R}^2. They are closed under their reflections, span EE, and are reduced — but the Cartan integers fail (R4): two adjacent vectors at 45∘45^\circ give 4cos⁡245∘=24\cos^2 45^\circ = 2, yet for vectors at 135∘135^\circ you'd need a length ratio incompatible with the octagon's equal lengths. No consistent integer Cartan data exists. This is precisely the obstruction that bans diagrams with a "cycle" of mirrors: the dihedral group I2(8)I_2(8) is a perfectly good Coxeter group but not a Weyl group — it is non-crystallographic. Likewise the icosahedral group H3H_3 and H4H_4, and I2(5)I_2(5) (fivefold symmetry), are Coxeter but never crystallographic, by the crystallographic restriction 4cos⁡2θ∈Z4\cos^2\theta\in\mathbb{Z}.

Connections

connection

Root systems sit at a crossroads. Via the Serre relations they reconstruct the entire Semisimple Lie Algebra from its Cartan matrix, making the classification of root systems identical to the classification of complex simple Lie algebras and (over algebraically closed fields) Linear Algebraic Groups. Forgetting the arrows yields Coxeter Groups, of which Weyl groups are the crystallographic members; this links to the geometry of Reflection Groups and hyperplane arrangements. Replacing positive-definiteness by positive-semidefinite or indefinite Cartan matrices yields affine and Kac–Moody algebras, generalizing [7] to infinite families. The simply-laced (ADEADE) diagrams reappear, astonishingly, as the McKay Correspondence (finite subgroups of SU(2)SU(2)), as Kleinian (du Val) surface singularities, and as quivers of finite representation type (Gabriel's theorem) — three independent classifications that all produce exactly An,Dn,E6,E7,E8A_n, D_n, E_6, E_7, E_8.

Real-World Applications

🔬 Where this appears

Historical Development

⏳ History
- 1873 — Sophus Lie founds the theory of continuous transformation groups, supplying the objects to be classified.
- 1888–1890 — Wilhelm Killing publishes four papers classifying simple complex Lie algebras, inventing roots, the rank, the Cartan matrix, and discovering G2,F4,E6,E7,E8G_2,F_4,E_6,E_7,E_8 (though with errors and tentative claims).
- 1894 — Élie Cartan's doctoral thesis corrects and rigorizes the classification, proves existence of the exceptional algebras, and establishes the root-system framework cleanly.
- 1925–1926 — Hermann Weyl develops the global theory: compact groups, the unitarian trick, Weyl group, and the Weyl character/integral formulas, anchoring representations to root data.
- 1934 — Coxeter classifies finite reflection groups and introduces Coxeter diagrams, separating angle data from crystallographic length data.
- 1947 — Eugene Dynkin introduces the diagrams now bearing his name, streamlining the classification into a few combinatorial lemmas.
- 1950s–1960s — Chevalley constructs the simple algebraic groups over arbitrary fields from root data ("Chevalley bases"), and the classification becomes scheme-theoretic.
- 1968 — Gabriel's theorem: quivers of finite representation type are exactly the ADE Dynkin diagrams, opening the modern web of ADE correspondences.
- 1980 — McKay's correspondence links finite subgroups of SU(2)SU(2) to affine ADE diagrams, deepening the unification of seemingly unrelated classifications.

Subtleties & Limits

open-problem

The classification of finite root systems is complete and clean, but the surrounding landscape harbors deep open questions. (i) Conceptual explanation of E8E_8: there is still no fully satisfying "reason" why the exceptional series exists; Vinberg, Deligne, and others have proposed exceptional-series interpolations (the "magic triangle," Deligne's exceptional series with uniform dimension formulas), yet a single unifying principle remains elusive. (ii) Kac–Moody and Borcherds algebras: dropping positive-definiteness to indefinite Cartan matrices yields hyperbolic and Lorentzian Kac–Moody algebras (e.g. E10,E11E_{10}, E_{11} conjecturally governing M-theory/supergravity); the structure of their root multiplicities is largely unknown, with no closed formula. (iii) Non-crystallographic to crystallographic: understanding precisely how H3,H4,I2(n)H_3,H_4,I_2(n) embed into and "fold from" crystallographic systems (Coxeter folding, quasicrystal projection) continues to generate research. (iv) Categorification: realizing root systems and their positive roots intrinsically as shadows of categorical/homological data (cluster categories, stability conditions, Bridgeland spaces) is an active program whose general structure is only partially understood.