Affine root system
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 and .Definition
Classification
The affine roots systems A1 = B1 = B = C1 = C are the same, as are the pairs B2 = C2, B = C, 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 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 | 2 if n=1, 1 if n≥2 | ,,,,... |
| Bn | 2 | ,,,... |
| B | 2 | ,,, ... |
| Cn | 3 | ,,,... |
| C | 3 | ,,,... |
| BCn | 2 if n=1, 3 if n ≥ 2 | ,,,,... |
| Dn | 1 | ,,,... |
| E6 | 1 | |
| E7 | 1 | |
| E8 | 1 | |
| F4 | 2 | |
| F | 2 | |
| G2 | 2 | |
| G | 2 | |
| | 3 if n=1, 4 if n≥2 | ,,,,... |
| | 3 if n=1, 4 if n≥2 | ,,,,... |
| | 4 if n=2, 3 if n≥3 | ,,,,... |
| | 4 if n=1, 5 if n≥2 | ,,,,... |
Applications
