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 the Cartan integers nβ,α = 2(β, α)/(α, α) are integers. Here that limits the angles between roots, and a root system shrinks to a few simple roots and a graph.
Take two linearly independent roots α and β at angle θ. Both Cartan integers
are integers, and their product is
Since 0 ≤ cos2θ < 1, the product is 0, 1, 2 or 3, so θ is 90°, 60° or 120°, 45° or 135°, or 30° or 150°. A product of 2 or 3 splits as 1 × 2 or 1 × 3, which fixes the ratio of squared lengths at 2 or 3.
Figure 1. Two vectors α and β with adjustable angle and squared-length ratio, and their two Cartan numbers, which must both be integers. A product of 2 or 3 needs a length ratio of 2 or 3, or its inverse: try 135° at ratio 2.
Finiteness alone would not force these angles. Closing α and β under reflection in each other gives a finite set at every whole-degree angle, the vectors of a regular polygon; what fails is that they stop being integer combinations of α and β.
Figure 2. α and β closed under reflection in each other, over the lattice ℤα + ℤβ (dots); dashed vectors are off the lattice. At 108° the closure is a regular pentagon's vectors. The strip covers every whole-degree angle; filled dots keep every vector on the lattice.
Only the pairs that Figure 1 allows keep the whole closure on the lattice. This is the crystallographic restriction: the pentagon's reflections form a finite group, but not the Weyl group of any Lie algebra.
Pick a linear functional h on 𝔼 that vanishes on no 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, and every positive root is a combination of them with nonnegative integer coefficients. Another h can give another Π, related to the first by a Weyl group element.
Distinct simple roots satisfy (α, β) ≤ 0, since otherwise α − β would be a root and would split α or β into positive roots. 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. Since
Aij is −2 or −3 exactly when αi is long and αj short. The Dynkin diagram has a node per simple root, AijAji lines between nodes i and j, and on a double or triple bond an arrow toward the short root. A disconnected diagram is a direct sum.
In rank 2, 90° gives A1 × A1, which is not irreducible, and 120°, 135° and 150° give A2, B2 and G2. C2 is B2 turned by 45° and rescaled, so B and C are listed separately only from rank 3.
Figure 3. The irreducible rank-2 root systems, B2 also in its C labelling. The functional h (arrow) picks the positive roots; the simple ones are highlighted, and the Cartan matrix and diagram are read off them. Turning h changes Π but never the matrix.
For E8 the 240 roots collapse to eight simple roots, each of squared length 2 in Bourbaki's coordinates; α1 alone has half-integer entries.
| x₁ | x₂ | x₃ | x₄ | x₅ | x₆ | x₇ | x₈ |
|---|
Figure 4. 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 5. The 8×8 Cartan matrix of E8, computed from the inner products of the simple roots above. Its fourteen −1 entries, seven above the diagonal, mark the seven pairs of simple roots at 120°, the edges of the Dynkin diagram in Figure 7.
Every root is an integer combination of the αi with all coefficients of one sign, and the coefficient sum is its height.
Figure 6. The 240 roots of E8, each solved for its coefficients on the simple roots. The selected one is the highest root. Click any 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 7. The E8 Dynkin diagram. The branch node α4 has legs of 1, 2 and 4 nodes. Deleting a simple root leaves a subsystem whose roots are filtered from the 240: α8 leaves E7, α1 leaves D7, and α7 leaves E6 ⊕ A1.
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 8. The classical families. The bounds on n avoid repeats at small rank, where B1 = C1 = A1, C2 = B2, D3 = A3 and D2 = A1 × A1.
Bn and Cn share dimension and Weyl group but differ from rank 3 on: the arrow records which end carries the short roots. Five exceptional diagrams complete the list: G2, F4, E6, E7 and E8.
The diagram is enough to rebuild everything else, with no coordinates. Starting from the simple roots: if αi can be subtracted from a root β exactly p times, it can be added p − Σj bjAji times, where the bj are β's coefficients.
Figure 9. Positive roots rebuilt from the Cartan matrix alone, one cell per root, stacked by height: simple roots blue, highest root purple. Rank plus twice the count is the dimension. For E8 the row lengths at the right return in Part 8 as its exponents.
Part 4 shows that these nine are the only connected graphs that can be Dynkin diagrams.