Part 5 of 15. From a root system Δ, pick a half-space, extract a minimal generating set of "simple" roots, record their pairwise Cartan integers in a matrix, and draw that matrix as a graph. Every simple Lie algebra compresses to a handful of nodes and edges, and the diagram for E8 already foreshadows the end of the story.
The last explainer reduced every pairwise root interaction to one of four types. We now compress an entire root system into a finite combinatorial signature.
Three steps: split Δ into positive and negative halves with a generic hyperplane; extract simple roots Π ⊂ Δ⁺, a minimal linearly independent generating set; record the pairwise Cartan integers among Π in a Cartan matrix, drawn as a graph called the Dynkin diagram. Each simple Lie algebra ends up identified with a small graph, and the classification reduces to asking which graphs are possible.
Pick any linear functional h: 𝔼 → ℝ nonzero on every root (a generic choice). Call α ∈ Δ positive if h(α) > 0, negative otherwise. This partitions Δ:
of equal size |Δ|/2, with Δ⁻ = −Δ⁺. Different h's give different splits, but any two differ by a Weyl group action, so derived quantities are Weyl-invariant and intrinsic to 𝔤.
A positive root α is simple (or indecomposable) if it cannot be written as α = β + γ for two positive roots β, γ. The set of simple roots is Π:
Three non-obvious facts: |Π| equals the rank n of 𝔤; Π is linearly independent, hence a basis for 𝔥*; and
every positive root is a nonnegative integer combination of simple roots, with unique coefficients. This generation property makes Π genuinely minimal.
Simple roots are at obtuse or right angles: (α, β) ≤ 0 for distinct α, β ∈ Π. Otherwise α − β would be a root of smaller height, contradicting indecomposability. Combined with the four legal angles, the angle between distinct simple roots is exactly one of 90°, 120°, 135°, 150°.
Given the simple roots Π = {α1, …, αn}, the Cartan matrix A is the n × n integer matrix whose entries are the pairwise Cartan integers:
Diagonal entries are 2. Off-diagonals Aij lie in {0, −1, −2, −3}. The matrix is "almost symmetric":
So same-length roots give Aij = Aji; different lengths give Aij / Aji equal to the squared-length ratio. The "more negative" entry corresponds to the shorter root.
A test for whether an integer matrix is a Cartan matrix of a simple Lie algebra: it must be positive definite with the right off-diagonal pattern. Positive definiteness is the heart of the classification, forcing the Dynkin diagram to be a connected tree of valency ≤ 3 with at most one multiple edge.
Eugene Dynkin's 1947 observation: all the Cartan matrix information fits in a small graph with one node per simple root and an edge between nodes i, j weighted by Aij · Aji ∈ {0, 1, 2, 3}. That graph is the Dynkin diagram:
Double and triple edges carry an arrow from long root to short root, breaking the length-induced asymmetry. With the arrow, the diagram encodes the full Cartan matrix losslessly.
A simple Lie algebra has an irreducible root system, so its Dynkin diagram is connected. Disconnected diagrams correspond to semisimple but non-simple algebras (direct sums). The classification problem becomes: which connected Dynkin diagrams have positive-definite Cartan matrices?
In rank 2 the Cartan matrix is 2 × 2 and there are four connected root systems (one per allowed angle): A2, B2, C2, G2. B2 and C2 are isomorphic as root systems but their Dynkin diagrams differ in arrow direction.
Figure 1. The four rank-2 root systems and their signatures. Each row pairs a root system (left) with its Cartan matrix and Dynkin diagram (right). The two simple roots α1 and α2 are highlighted in blue; the other roots are coloured by positivity. Notice that A2 and B2/C2 differ only in where the arrow points; G2 is the most extreme case with a 150° angle and a triple edge.
E8 has 240 roots in 8 dimensions. The Dynkin procedure collapses them to 8 simple roots, then to a graph of 8 nodes and 7 edges. Every property of E8 can be reconstructed from that diagram.
The Bourbaki simple roots are eight vectors in ℝ⁸, all of squared length 2. α1 is the only half-integer vector; the others are D8-type integer vectors.
| x₁ | x₂ | x₃ | x₄ | x₅ | x₆ | x₇ | x₈ |
|---|
Figure 2. The 8 Bourbaki simple roots of E8. These are the building blocks every other root is written against.
Because every simple root has squared length 2, the Cartan entries simplify to . Each diagonal entry is 2; each off-diagonal entry is 0 (orthogonal) or −1 (120°).
Figure 3. The 8×8 Cartan matrix of E8 in Bourbaki order. Diagonal entries (blue) are all 2. The seven −1 entries (red) encode the seven pairs of simple roots at 120° — these become the edges of the Dynkin diagram.
Figure 4. The E8 Dynkin diagram. Eight nodes, seven edges. Node α4 (red) is the branch node: three neighbours. The three "legs" emanating from α4 have edge-lengths 1, 2, 4 — the signature pattern that makes this E8 and not some other exceptional diagram.
Why legs of length 1, 2, 4? Because that is the maximum leg length whose Cartan matrix stays positive definite. Extending the long leg to 5 produces a zero eigenvalue, which is the affine Kac-Moody regime. E8 sits on the last finite rung.
Every one of the 240 roots decomposes uniquely as an integer combination of the αi, with coefficients all of one sign. The coefficient sum is the height; the highest root sits at height 29 with coefficients (2, 3, 4, 6, 5, 4, 3, 2).
Figure 5. The 240 roots of E8, each written in the simple-root basis. Click any root to see its coefficient bar chart; no root has mixed signs. Try finding the highest root — the single bar chart that reaches (2, 3, 4, 6, 5, 4, 3, 2).
What happens if we delete a node? The remaining 7 nodes form a smaller Dynkin diagram, a rank-7 root subsystem of E8 determined by which node is removed.
Figure 6. Deleting a single simple root from the E8 diagram. The surviving sub-diagram identifies the root subsystem: E7 (delete α8), D7 (delete α1), E6 ⊕ A1 (delete α7), and so on. This "peeling" is the operation that builds every exceptional subsystem we will meet in Act III.
Applied to the rank-n classical families An, Bn, Cn, Dn, the construction produces a distinctive signature for each:
| Type | Rank | Dynkin diagram shape | Signature |
|---|---|---|---|
| An | n ≥ 1 | linear chain of n nodes, all single edges | "the straightforward type", no multi-edges, no branches |
| Bn | n ≥ 2 | chain of n−1 nodes + 1 node attached at the end by a double edge with arrow pointing away from the chain | "long-root chain ending in a short root" |
| Cn | n ≥ 2 | chain of n−1 nodes + 1 node attached at the end by a double edge with arrow pointing toward the chain | "short-root chain ending in a long root", Bn's reverse |
| Dn | n ≥ 3 | main chain of n−1 nodes with one extra node branching off the penultimate (αn−2) node | "the forked type", one Y-shaped branch, all single edges |
Figure 7. The Dynkin-diagram signatures of the four classical families. All four are trees (no loops). An and Dn are simply-laced, only single edges, all roots the same length. Bn and Cn each have exactly one double edge, placed at the end of the chain. The differences are small but consequential: the rank-2 cases collapse, but starting from rank 3 the four families are genuinely distinct.
Once the classification is complete, the only additional Dynkin diagrams are the five exceptional ones G2, F4, E6, E7, E8. Each gets its own chapter in Act III; the nine families form the complete Cartan–Killing list.
Every simple Lie algebra over ℂ is determined up to isomorphism by its Dynkin diagram alone. The Lie algebra plus all its representation theory and Weyl group reduce to a small graph.
What remains is to characterise which graphs can be valid Dynkin diagrams, equivalently which integer matrices of the right form can be positive definite. The next two explainers settle this with a short graph-theoretic argument.
The endgame. Classifying simple Lie algebras over ℂ is equivalent to classifying connected Dynkin diagrams with positive-definite Cartan matrices.