Part 3 of 9. Integrality allows only four angles between roots. Simple roots and their Cartan matrix then compress a whole root system into a small graph, the Dynkin diagram.
Part 2 showed that nβ,α = 2(β, α)/(α, α) is an integer for any two roots α, β. This part uses integrality to limit the angles between roots, then reduces a root system to a basis of simple roots and a small graph that records their angles and lengths.
Take two linearly independent roots α and β with angle θ between them. Both Cartan integers
are integers, and their product is
Since 0 ≤ cos2θ < 1 for independent roots, the product is an integer in {0, 1, 2, 3}. Each value pins down the geometry:
When the lengths differ, the larger integer is the one that divides by the shorter root: if |β| > |α| then |nβ,α| = 2|β| |cos θ| / |α| is the bigger of the two. For B2, with a long root β = ε1 − ε2 and a short root α = ε2, nβ,α = 2(−1)/1 = −2 and nα,β = 2(−1)/2 = −1.
Figure 1. Two vectors α and β with adjustable angle and squared-length ratio, and their two Cartan numbers. A configuration can occur in a root system only if both are integers. Try 135° with ratio 2 and 150° with ratio 3, the only ways to get a product of 2 or 3.
Pick a linear functional h on 𝔼 that is nonzero on every root, and call a root positive when h(α) > 0:
A positive root is simple if it is not the sum of two positive roots:
The simple roots form a basis of 𝔼, and every positive root is a combination of them with nonnegative integer coefficients. Different choices of h give different Π, but any two are related by an element of the Weyl group, so nothing derived from Π depends on the choice.
Distinct simple roots satisfy (α, β) ≤ 0. If the inner product were positive, α − β would be a root, and one of α = β + (α − β) or β = α + (β − α) would decompose a simple root. So between simple roots only 90°, 120°, 135° and 150° survive.
For simple roots α1, …, αn, the Cartan matrix collects the pairwise integers:
The diagonal is 2 and each off-diagonal entry is 0, −1, −2 or −3. The matrix is symmetric up to root lengths:
so Aij = −2 or −3 exactly when αi is the long root and αj the short one.
Dynkin's diagram draws the same data as a graph: one node per simple root, and AijAji ∈ {0, 1, 2, 3} lines between nodes i and j. A double or triple bond carries an arrow from the long root to the short one, recording which entry is larger. With the arrow the diagram determines the Cartan matrix, which determines the Lie algebra. A simple algebra has a connected diagram; a disconnected diagram is a direct sum.
In rank 2 the four possible angles between the simple roots give four root systems. At 90° the two roots ignore each other and the system is A1 × A1, which is not irreducible. The other three angles give A2, with a single bond, B2, with a double bond, and G2, with a triple bond. The figure also shows C2, but B2 and C2 are the same root system, rotated by 45° and rescaled, which is why the two families are listed separately only from rank 3 on.
Figure 2. The irreducible rank-2 root systems, with B2 shown in both its B and C labellings, generated from their highlighted simple roots, with the Cartan matrix computed from inner products and the Dynkin diagram read off the matrix.
The same procedure collapses the 240 roots of E8 to eight simple roots and a graph with eight nodes and seven edges. In Bourbaki's coordinates every simple root has squared length 2. α1 is the only one with half-integer entries; the rest are integer vectors of the form εi ± εj.
| x₁ | x₂ | x₃ | x₄ | x₅ | x₆ | x₇ | x₈ |
|---|
Figure 3. The eight Bourbaki simple roots of E8, read from the shared E8 library.
With every squared length 2, the Cartan entries reduce to : 2 on the diagonal, and 0 or −1 off it.
Figure 4. The 8×8 Cartan matrix of E8, computed from the inner products of the simple roots above. The seven −1 entries mark the seven pairs of simple roots at 120°, which become the edges of the Dynkin diagram in Figure 6.
Every root is an integer combination of the αi with all coefficients of one sign. The coefficient sum is the height. The highest root has height 29 and coefficients (2, 3, 4, 6, 5, 4, 3, 2) in Bourbaki order.
Figure 5. The 240 roots of E8, each solved for its coordinates in the simple-root basis. Click a root to see its coefficients. No root has mixed signs.
Deleting a node leaves a smaller diagram, which names a rank-7 root subsystem of E8.
Figure 6. The E8 Dynkin diagram. Node α4 is the branch node, with three neighbours, and the three legs leaving it have 1, 2 and 4 nodes. Deleting one simple root leaves a smaller diagram. Removing α8 leaves E7, α1 leaves D7, and α7 leaves E6 ⊕ A1. Each subsystem's root count comes from filtering the 240 roots.
The four infinite families come from the matrix algebras of Part 1, and their diagrams differ only at one end.
| Type | Algebra | Dimension | Diagram |
|---|---|---|---|
| An, n ≥ 1 | sln+1 | n(n + 2) | a chain of n nodes, single bonds |
| Bn, n ≥ 2 | so2n+1 | n(2n + 1) | a chain whose last bond is double, arrow pointing to the short end node εn |
| Cn, n ≥ 3 | sp2n | n(2n + 1) | the same chain with the arrow reversed: the end node 2εn is long |
| Dn, n ≥ 4 | so2n | n(2n − 1) | a chain of n − 2 nodes that forks into two at one end |
Figure 7. The classical families. The lower bounds on n avoid repeats: at small rank the diagrams coincide, giving B1 = C1 = A1 (so3 ≅ sp2 ≅ sl2), C2 = B2 (sp4 ≅ so5), D3 = A3 (so6 ≅ sl4), and D2 = A1 × A1 (so4 ≅ sl2 ⊕ sl2).
Bn and Cn have the same dimension and the same Weyl group for every n, yet from rank 3 on they are different algebras: the arrow records which end of the chain carries the short roots. Beyond these four families there are five more diagrams: G2, F4, E6, E7 and E8. Part 4 shows that these nine are the only connected graphs that can be Dynkin diagrams.