In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebras and superalgebras, and semisimple p-adic algebraic groups, and correspond to families of Macdonald polynomials. The reduced affine root systems were used by Kac and Moody in their work on Kac–Moody algebras. Possibly non-reduced affine root systems were introduced and classified by Macdonald (1972) and Bruhat & Tits (1972) (except that both these papers accidentally omitted the Dynkin diagram ).
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The affine roots systems A1 = B1 = B∨
1 = C1 = C∨
1 are the same, as are the pairs B2 = C2, B∨
2 = C∨
2, and A3 = D3
The number of orbits given in the table is the number of orbits of simple roots under the Weyl group. In the Dynkin diagrams, the non-reduced simple roots α (with 2α a root) are colored green. The first Dynkin diagram in a series sometimes does not follow the same rule as the others.
| Affine root system | Number of orbits | Dynkin diagram |
|---|---|---|
| An (n ≥ 1) | 2 if n=1, 1 if n≥2 | , , , , ... |
| Bn (n ≥ 3) | 2 | , ,, ... |
| B∨ n (n ≥ 3) |
2 | , ,, ... |
| Cn (n ≥ 2) | 3 | , , , ... |
| C∨ n (n ≥ 2) |
3 | , , , ... |
| BCn (n ≥ 1) | 2 if n=1, 3 if n ≥ 2 | , , , , ... |
| Dn (n ≥ 4) | 1 | , , , ... |
| E6 | 1 | |
| E7 | 1 | |
| E8 | 1 | |
| F4 | 2 | |
| F∨ 4 |
2 | |
| G2 | 2 | |
| G∨ 2 |
2 | |
| (BCn, Cn) (n ≥ 1) | 3 if n=1, 4 if n≥2 | , , , , ... |
| (C∨ n, BCn) (n ≥ 1) |
3 if n=1, 4 if n≥2 | , , , , ... |
| (Bn, B∨ n) (n ≥ 2) |
4 if n=2, 3 if n≥3 | , , ,, ... |
| (C∨ n, Cn) (n ≥ 1) |
4 if n=1, 5 if n≥2 | , , , , ... |