07.04.01 · representation-theory / classification

Cartan-Weyl classification

shipped3 tiersLean: partial

Anchor (Master): Humphreys; Bourbaki Ch. IV–VI; Knapp; Helgason

Intuition [Beginner]

The Cartan-Killing classification (also called Cartan-Weyl classification) is one of the great triumphs of 19th- and 20th-century mathematics: a complete list of all simple complex Lie algebras, expressed through Dynkin diagrams — beautiful little graphs with at most 8 nodes that capture the entire structure.

There are exactly four infinite families and five exceptional cases:

$A_{n}$ : $sl_{n + 1}$ , traceless matrices.
$B_{n}$ : $so_{2 n + 1}$ , antisymmetric matrices in odd dimension.
$C_{n}$ : $sp_{2 n}$ , the symplectic algebra.
$D_{n}$ : $so_{2 n}$ , antisymmetric matrices in even dimension.
Exceptionals: $G_{2}, F_{4}, E_{6}, E_{7}, E_{8}$ — five sporadic simple algebras with no infinite-family pattern.

Each Dynkin diagram is a small finite graph with edges of specific multiplicities. From this graph alone, the entire structure of the corresponding Lie algebra is determined: dimension, root system, representations, Weyl group.

This classification is a finite answer to an infinite problem: from the abstract notion "simple complex Lie algebra," the rigid combinatorial structure forces only these specific possibilities.

Visual [Beginner]

The five exceptional Dynkin diagrams: $G_{2}, F_{4}, E_{6}, E_{7}, E_{8}$ , alongside the four infinite families $A_{n}, B_{n}, C_{n}, D_{n}$ .

Worked example [Beginner]

The Lie algebra $sl_{3} (C)$ corresponds to the Dynkin diagram $A_{2}$ : two nodes connected by a single edge. From this 2-node diagram alone:

Rank (number of nodes) = 2: there is a 2-dimensional Cartan subalgebra.
Number of positive roots = 3 (read off from the structure of $A_{2}$ ).
Total dimension = rank + 2 × (positive roots) = 2 + 2 × 3 = 8.
Weyl group = symmetric group $S_{3}$ of order 6.

So $dim sl_{3} = 8$ . (Indeed, $sl_{3} = {3 \times 3 traceless matrices}$ , dimension $3^{2} - 1 = 8$ . ✓)

The exceptional algebra $E_{8}$ has rank 8 and 240 positive roots; its dimension is $8 + 2 \cdot 120 = 248$ . The smallest faithful representation of $E_{8}$ is 248-dimensional (the adjoint representation itself). The structure of $E_{8}$ — its root system, weight lattice, and Weyl group — is encoded in its 8-node Dynkin diagram.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $g$ be a complex semisimple Lie algebra with Cartan subalgebra $h \subseteq g$ . The root system $Φ \subset h^{*}$ is the set of nonzero $h$ -weights in the adjoint representation:

g_{α} := {X \in g : [H, X] = α (H) X for all H \in h}, Φ = {α \neq = 0 : g_{α} \neq = 0} .

Axioms of an abstract root system. A subset $Φ \subset V$ of a real Euclidean space $(V, ⟨ \cdot, \cdot ⟩)$ is a root system if:

(R1) $Φ$ is finite, spans $V$ , and $0 \in / Φ$ .

(R2) For each $α \in Φ$ , $R α \cap Φ = {\pm α}$ .

(R3) For each $α \in Φ$ , the reflection $s_{α} (β) := β - \frac{2 ⟨ β , α ⟩}{⟨ α , α ⟩} α$ permutes $Φ$ .

(R4) For all $α, β \in Φ$ , $\frac{2 ⟨ β , α ⟩}{⟨ α , α ⟩} \in Z$ (the crystallographic condition).

A root system is irreducible if it cannot be partitioned into two orthogonal subsystems (in disjoint subspaces). The rank is $dim V$ .

Theorem (Cartan-Killing classification of irreducible root systems). Every irreducible root system is isomorphic to one of:

$A_{n}$ for $n \geq 1$ , rank $n$ , $Φ = {e_{i} - e_{j} : i \neq = j}$ , $∣Φ∣ = n (n + 1)$ .
$B_{n}$ for $n \geq 2$ , rank $n$ , $Φ = {\pm e_{i}, \pm e_{i} \pm e_{j}}$ , $∣Φ∣ = 2 n^{2}$ .
$C_{n}$ for $n \geq 3$ , rank $n$ , $Φ = {\pm 2 e_{i}, \pm e_{i} \pm e_{j}}$ , $∣Φ∣ = 2 n^{2}$ .
$D_{n}$ for $n \geq 4$ , rank $n$ , $Φ = {\pm e_{i} \pm e_{j}}$ , $∣Φ∣ = 2 n (n - 1)$ .
Exceptionals: $G_{2}$ (rank 2, $∣Φ∣ = 12$ ), $F_{4}$ (rank 4, $∣Φ∣ = 48$ ), $E_{6}$ (rank 6, $∣Φ∣ = 72$ ), $E_{7}$ (rank 7, $∣Φ∣ = 126$ ), $E_{8}$ (rank 8, $∣Φ∣ = 240$ ).

Dynkin diagrams. Each irreducible root system has a basis $Δ = {α_{1}, \dots, α_{n}}$ of simple roots (a basis of $V$ such that every root is a non-negative or non-positive integer combination). The Dynkin diagram is a graph:

Vertices = simple roots $α_{1}, \dots, α_{n}$ .
Edge between $α_{i}$ and $α_{j}$ with multiplicity $\frac{4 ⟨ α _{i} , α _{j} ⟩ ^{2}}{⟨ α _{i} , α _{i} ⟩ ⟨ α _{j} , α _{j} ⟩} \in {0, 1, 2, 3}$ , with an arrow pointing toward the shorter root.

Possible Dynkin diagrams: simply-laced ( $A_{n}, D_{n}, E_{6}, E_{7}, E_{8}$ ), with double edges ( $B_{n}, C_{n}, F_{4}$ ), with triple edges ( $G_{2}$ ). The complete list comprises the 4 infinite families and the 5 exceptionals.

From Dynkin diagram to Lie algebra. Given a Dynkin diagram, the Serre relations uniquely reconstruct the simple complex Lie algebra:

[E_{i}, F_{j}] = δ_{ij} H_{i}, [H_{i}, E_{j}] = a_{ij} E_{j}, [H_{i}, F_{j}] = - a_{ij} F_{j}, (ad E_{i})^{1 - a_{ij}} E_{j} = 0, (ad F_{i})^{1 - a_{ij}} F_{j} = 0,

where $a_{ij} = \frac{2 ⟨ α _{i} , α _{j} ⟩}{⟨ α _{i} , α _{i} ⟩}$ is the Cartan matrix (encoded in the Dynkin diagram).

Compact real forms. Every simple complex Lie algebra has a unique compact real form $g_{0}$ (with negative-definite Killing form). The classification of compact real simple Lie groups is essentially the same: $SU (n + 1)$ for $A_{n}$ , $SO (2 n + 1)$ for $B_{n}$ , $Sp (n)$ for $C_{n}$ , $SO (2 n)$ for $D_{n}$ , and the compact forms of the 5 exceptional algebras.

Key theorem with proof [Intermediate+]

Theorem (Cartan-Killing classification, modern statement). The map

{irreducible Dynkin diagrams} ⟷ {simple complex Lie algebras} / ≅

is a bijection.

Proof sketch.

Step 1 (Lie algebra → root system → Dynkin diagram). Given a simple complex Lie algebra $g$ , choose a Cartan subalgebra $h$ (any two are conjugate by inner automorphisms), and form the root system $Φ \subset h^{*}$ from the adjoint representation. The Killing form $κ$ provides a positive-definite pairing on the real span of $Φ$ . Choose a base $Δ \subset Φ$ and form the Dynkin diagram from inner products of simple roots. Different choices give isomorphic Dynkin diagrams (Weyl group acts transitively on bases, and the Dynkin diagram is invariant).

Step 2 (Classification of irreducible root systems). The crystallographic condition $\frac{2 ⟨ α , β ⟩}{⟨ α , α ⟩} \in Z$ severely restricts angles between pairs of roots: only $90°, 60°/120°, 45°/135°, 30°/150°$ are possible. By careful combinatorial analysis (going back to Killing 1888, polished by Cartan), the only possible irreducible root systems are $A_{n}, B_{n}, C_{n}, D_{n}, G_{2}, F_{4}, E_{6}, E_{7}, E_{8}$ . Each diagram type is checked to actually exist (constructions for each).

Step 3 (Dynkin diagram → Lie algebra). Given a Dynkin diagram, the Serre relations construct a Lie algebra freely on generators ${E_{i}, F_{i}, H_{i}}$ subject to the relations encoded by the Cartan matrix. Serre's theorem shows this is finite-dimensional and simple, with the prescribed Dynkin diagram.

So the map is surjective and well-defined. Injectivity follows from the rigidity of the structure: two simple complex Lie algebras with isomorphic Dynkin diagrams are isomorphic. $□$

The classification was proved in full by Wilhelm Killing (1888-90), with errors and gaps; Élie Cartan (1894) provided rigorous proofs and unified treatment; later refined by Eugene Dynkin (1947) introducing the diagrams that bear his name, and by Jean-Pierre Serre (1966) with the Serre relations.

Bridge. The construction here builds toward later units of the strand, where the same pattern is taken up at higher structure. The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Exercise 3 (medium, proof).

Show that the angle between two distinct simple roots in any root system is one of: $90°, 120°, 135°, 150°$ .

Hint

Use the crystallographic condition and the Cauchy-Schwarz inequality.

Answer

Let $α, β$ be distinct simple roots. Define $a := \frac{2 ⟨ α , β ⟩}{⟨ α , α ⟩}, b := \frac{2 ⟨ β , α ⟩}{⟨ β , β ⟩}$ , both integers. Then

ab = \frac{4 ⟨ α , β ⟩ ^{2}}{⟨ α , α ⟩ ⟨ β , β ⟩} = 4 cos^{2} (θ) \in [0, 4),

with equality 4 only if $α = \pm β$ (excluded since $α \neq = \pm β$ as simple roots). So $ab \in {0, 1, 2, 3}$ .

For simple roots, the inner product $⟨ α, β ⟩ \leq 0$ (foundational fact). So both $a$ and $b$ are non-positive. Hence $ab \in {0, 1, 2, 3}$ and both $a, b \leq 0$ , giving $cos (θ) \in {0, - 1/2, - 2 /2, - 3 /2}$ , i.e., $θ \in {90°, 120°, 135°, 150°}$ . $□$

These angles correspond to Dynkin diagram edges: 90° = no edge, 120° = single edge ( $A$ / $D$ / $E$ ), 135° = double edge ( $B$ / $C$ / $F$ ), 150° = triple edge ( $G$ ).

Exercise 5 (medium, conceptual).

Why does the Cartan-Weyl classification go through root systems rather than directly through Lie algebras?

Hint

Root systems are simple combinatorial objects living in finite-dimensional Euclidean space.

Answer

Root systems $Φ \subset V$ in finite-dimensional Euclidean space are combinatorial objects: finite sets of vectors with specific reflection-symmetry and integrality properties. The crystallographic condition severely restricts the possibilities, and the entire theory becomes a problem in classical (finite!) Euclidean geometry — far more tractable than abstract Lie algebra theory.

The brilliance of Killing/Cartan was to reduce the abstract Lie algebra problem to the root-system problem. Once you know the irreducible root systems are exactly $A_{n}, B_{n}, C_{n}, D_{n}, G_{2}, F_{4}, E_{6}, E_{7}, E_{8}$ , you've classified the Lie algebras, because:

Each Lie algebra determines a root system (via Cartan + adjoint representation).
Each root system determines a Lie algebra (via Serre relations / Chevalley basis).

The root system serves as a combinatorial fingerprint unique to each isomorphism class of simple Lie algebra. This pattern — combinatorial fingerprint via simple root data — recurs throughout 20th-century mathematics (Coxeter groups, buildings, Tits geometries, Kac-Moody algebras).

Exercise 6 (hard, proof).

Outline the proof that any rank-2 irreducible root system is one of $A_{2}, B_{2} = C_{2}, G_{2}$ .

Hint

Use the angle restrictions from Exercise 3.

Answer

For rank 2, choose simple roots $α, β$ with $∣ α ∣ \leq ∣ β ∣$ . The angle is one of $90°, 120°, 135°, 150°$ .

90°: $α ⊥ β$ . The system splits as ${\pm α} \cup {\pm β}$ , not irreducible.
120° with $∣ α ∣ = ∣ β ∣$ : this is $A_{2}$ . Roots: ${\pm α, \pm β, \pm (α + β)}$ , total 6.
135° with $∣ β ∣^{2} = 2∣ α ∣^{2}$ : this is $B_{2} = C_{2}$ . Roots: ${\pm α, \pm β, \pm (α + β), \pm (2 α + β)}$ , total 8.
150° with $∣ β ∣^{2} = 3∣ α ∣^{2}$ : this is $G_{2}$ . Roots: 12 total, with 6 short and 6 long.

So the rank-2 irreducible root systems are exactly $A_{2}, B_{2}, G_{2}$ (with $B_{2} = C_{2}$ as abstract diagrams). $□$

The full higher-rank classification follows similar but more elaborate combinatorial reasoning, with the exceptional cases $F_{4}, E_{6}, E_{7}, E_{8}$ requiring careful diagram analysis.

Exercise 7 (hard, conceptual).

State the Langlands dual of an irreducible root system and explain its significance.

Hint

Swap roots and coroots; certain types are self-dual, certain types swap.

Answer

For an irreducible root system $Φ$ , define the coroots $α^{\lor} := \frac{2 α}{⟨ α , α ⟩}$ for each $α \in Φ$ . The dual root system $Φ^{\lor}$ consists of all coroots and has the same axiomatic structure as $Φ$ .

The duality on Dynkin diagram types:

$A_{n}^{\lor} = A_{n}$ (self-dual).
$B_{n}^{\lor} = C_{n}$ (swap!) and $C_{n}^{\lor} = B_{n}$ .
$D_{n}^{\lor} = D_{n}$ (self-dual).
$E_{6}^{\lor} = E_{6}$ , $E_{7}^{\lor} = E_{7}$ , $E_{8}^{\lor} = E_{8}$ (all self-dual).
$G_{2}^{\lor} = G_{2}$ , $F_{4}^{\lor} = F_{4}$ (self-dual but with reversed arrows).

Significance. This is the classical "Langlands duality" between root systems. In the Langlands programme, automorphic representations of a reductive group $G$ are conjecturally dual to Galois representations of its Langlands dual $G^{\lor}$ . The dual root system $Φ^{\lor}$ governs the Lie algebra of $G^{\lor}$ . So $GL_{n}^{\lor} = GL_{n}$ but $Sp_{2 n}^{\lor} = SO_{2 n + 1}$ — a deep fact tied to Whittaker models, $L$ -functions, and number theory.

The Geometric Langlands programme conjectures a categorical analogue: $D$ -modules on $Bun_{G}$ correspond to quasi-coherent sheaves on $Loc_{G^{\lor}}$ , with the dual root system playing a fundamental role.

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has root systems and Cartan matrices but does not yet have the full classification theorem.

[object Promise]

Advanced results [Master]

Kac-Moody algebras. Generalisations of finite-dimensional simple Lie algebras to infinite-dimensional algebras with Cartan matrices that are not necessarily positive-definite. Three types:

Finite type: positive-definite Cartan matrix $\Leftrightarrow$ classical simple Lie algebras.
Affine type: positive-semidefinite (1-dim kernel) $\Leftrightarrow$ affine Lie algebras $\hat{g}$ .
Indefinite type: hyperbolic and beyond, including the Monster Lie algebra of Borcherds.

Building theory (Tits). Each simple Lie group has an associated building — a simplicial complex encoding its parabolic structure, with apartments isomorphic to Coxeter complexes of the Weyl group. Buildings classify Tits systems / BN-pairs and provide a geometric framework for the structure of reductive groups.

Twin buildings (Tits-Ronan). For Kac-Moody groups, twin buildings replace the classical building, providing a Lie-theoretic framework even in infinite dimensions.

Modular Lie algebras. Over fields of positive characteristic, the Cartan-Killing classification fails; new exceptional simple Lie algebras appear (Witt, Jacobson, Hamilton, Contact, plus Melikyan in characteristic 5). The full classification was completed in characteristic $p \geq 7$ by Premet-Strade.

Connection to ADE singularities (Brieskorn-Slodowy). Simply-laced Dynkin diagrams ( $A_{n}, D_{n}, E_{6}, E_{7}, E_{8}$ ) classify simple surface singularities and discrete subgroups of $SU (2)$ via the McKay correspondence. The exceptionality of $E_{8}$ also appears in lattice theory (the $E_{8}$ lattice is the densest sphere packing in 8 dimensions, by Viazovska 2016).

Connections to physics. $E_{8}$ appears in heterotic string theory (the gauge group is $E_{8} \times E_{8}$ ). The classification of supersymmetric quantum field theories includes the $E_{8}$ Higgs branch in M-theory. Garrett Lisi's "exceptionally simple theory of everything" (2007) attempted to unify all forces in $E_{8}$ — controversial but stimulating.

Synthesis. This construction generalises the pattern fixed in 07.01.01 (group representation), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

The full classification proof is given in Humphreys Introduction to Lie Algebras and Representation Theory §8–§12 (in 80 pages of careful exposition), and Bourbaki Lie Groups and Lie Algebras Ch. VI–VIII. The critical steps are: rigidity of Cartan subalgebras (any two are inner-conjugate), root system axioms from semisimplicity, classification of irreducible root systems via crystallographic condition, Serre relations reconstructing the Lie algebra from the Cartan matrix, finite-dimensionality of the constructed algebras.

Connections [Master]

Group representation 07.01.01 — the Cartan-Weyl classification organises representations of all simple Lie algebras and Lie groups.
Schur's lemma 07.01.02 — foundational input.
Highest weight representation 07.03.01 — the classification provides the structural framework on which highest-weight theory operates.
Lie algebra 03.04.01 — the abstract algebraic objects being classified.
Lie group 03.03.01 — Lie group classification follows from Lie algebra classification + global topological choices.
Spin group 03.09.03 — $Spin (n)$ groups are real forms of $B_{n}, D_{n}$ Lie algebras.
Orthogonal group 03.03.03 — $O (n)$ structures correspond to $B_{n}, D_{n}$ types.
Symplectic group 05.01.03 — $Sp (2 n)$ is type $C_{n}$ .
Topological K-theory 03.08.01 — classifying spaces $B G$ for simple compact Lie groups have rational cohomology generated by Pontryagin / Chern classes whose structure mirrors the root system.

Historical & philosophical context [Master]

Wilhelm Killing began the classification in the 1880s, motivated by Sophus Lie's earlier work on continuous groups of transformations. Killing's 1888-90 papers, Die Zusammensetzung der stetigen endlichen Transformationsgruppen, contained the four infinite families and the five exceptionals — though with errors and obscure proofs (Killing's prose was famously difficult).

Élie Cartan's 1894 dissertation Sur la structure des groupes de transformations finis et continus gave rigorous proofs and synthesised the theory. Cartan also classified all real simple Lie algebras (multiple real forms of each complex algebra) and computed many of their representations explicitly.

Hermann Weyl extended the theory in the 1920s, proving:

The Weyl character formula and Weyl dimension formula.
The unitarian trick: complex semisimple Lie groups have the same finite-dim rep theory as their compact real forms.
Compactness of the universal cover of compact semisimple Lie groups (extending Cartan's earlier results).
Weyl's rigidity theorem: deformations of compact Lie group representations are inert.

The 1947 paper of Eugene Dynkin introduced the Dynkin diagrams that bear his name, providing the visual language now universal. Jean-Pierre Serre's 1966 Algèbres de Lie semi-simples complexes gave the Serre relations reconstructing the Lie algebra directly from the Dynkin diagram.

The exceptional Lie algebras $G_{2}, F_{4}, E_{6}, E_{7}, E_{8}$ have been a continuing source of mystery and depth. They appear unexpectedly throughout mathematics and physics:

$G_{2}$ as the automorphism group of the octonions.
$F_{4}$ as the automorphism group of the exceptional Jordan algebra.
$E_{6}, E_{7}, E_{8}$ in the Freudenthal-Tits magic square of Lie groups built from division algebras.
$E_{8}$ as the gauge group of heterotic string theory and the densest 8-dimensional lattice.

The classification is not just a list — it is a window into the deep mathematical structure of symmetry, and into why symmetry has the specific forms it does. Few results in mathematics combine elegance, depth, and surprise like the Cartan-Weyl classification.

Bibliography [Master]

Humphreys, Introduction to Lie Algebras and Representation Theory — §8–§12 give the canonical classification proof.
Fulton & Harris, Representation Theory: A First Course — Lecture 21 sketches the classification.
Bourbaki, Lie Groups and Lie Algebras Ch. IV–VI — comprehensive structural treatment.
Knapp, Lie Groups Beyond an Introduction — Ch. II treats the structure of compact Lie groups.
Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces — the classification of real semisimple algebras and their associated symmetric spaces.
Kac, Infinite-Dimensional Lie Algebras — the Kac-Moody generalisation.
Tits, Buildings of Spherical Type and Finite BN-Pairs — the building-theoretic perspective.
Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen" (1888-90) — the original.
Cartan, Sur la structure des groupes de transformations finis et continus (1894) — the rigorous classification.
Dynkin, "Classification of simple Lie algebras" (1947) — diagrams.

Prerequisites

07.01.01
07.03.01
03.04.01

Tier anchors

beginner: Strogatz analogous level for the periodic table of simple Lie groups
intermediate: Humphreys §10–§12; Fulton-Harris §21; Bourbaki Ch. VI
master: Humphreys; Bourbaki Ch. IV–VI; Knapp; Helgason

References

TODO_REF
Humphreys — Introduction to Lie Algebras and Representation Theory · §10–§12, classification of root systems and simple Lie algebras
TODO_REF
Fulton-Harris — Representation Theory · Lecture 21, Cartan-Killing classification
TODO_REF
Bourbaki — Lie Groups and Lie Algebras Ch. IV–VI · Root systems, Coxeter groups, Dynkin diagrams
TODO_REF
Knapp — Lie Groups Beyond an Introduction · Ch. II, structure of compact Lie groups

Lean module

Codex.RepresentationTheory.Classification.CartanWeyl

Reviewer

TBD

Estimated time

beginner: 18m
intermediate: 50m
master: 75m