Affine root system
Encyclopedia
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 algebra
s and superalgebras, and semisimple p-adic algebraic group
s, 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 (except that both these papers accidentally omitted the Dynkin diagram ).
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.
used affine root systems to study p-adic algebraic groups.
Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...
of affine-linear functions on a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. They are used in the classification of affine Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s and superalgebras, and semisimple p-adic algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s, 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 (except that both these papers accidentally omitted the Dynkin diagram ).
Classification
The affine roots systems A1 = B1 = B = C1 = C are the same, as are the pairs B2 = C2, B = C, and A3 = D3The 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 ≥ 3) | 2 | , ,, ... |
| Cn (n ≥ 2) | 3 | , , , ... |
| C (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 | 2 | |
| G2 | 2 | |
| G | 2 | |
| (BCn, Cn) (n ≥ 1) | 3 if n=1, 4 if n≥2 | , , , , ... |
| (C, BCn) (n ≥ 1) | 3 if n=1, 4 if n≥2 | , , , , ... |
| (Bn, B) (n ≥ 2) | 4 if n=2, 3 if n≥3 | , , ,, ... |
| (C, Cn) (n ≥ 1) | 4 if n=1, 5 if n≥2 | , , , , ... |
Irreducible affine root systems by rank
- Rank 1: A1, BC1, (BC1, C1), (C, BC1), (C, C1).
- Rank 2: A2, C2, C, BC2, (BC2, C2), (C, BC2), (B2, B), (C, C2), G2, G.
- Rank 3: A3, B3, B, C3, C, BC3, (BC3, C3), (C, BC3), (B3, B), (C, C3).
- Rank 4: A4, B4, B, C4, C, BC4, (BC4, C4), (C, BC4), (B4, B), (C, C4), D4, F4, F.
- Rank 5: A5, B5, B, C5, C, BC5, (BC5, C5), (C, BC5), (B5, B), (C, C5), D5.
- Rank 6: A6, B6, B, C6, C, BC6, (BC6, C6), (C, BC6), (B6, B), (C, C6), D6, E6,
- Rank 7: A7, B7, B, C7, C, BC7, (BC7, C7), (C, BC7), (B7, B), (C, C7), D7, E7,
- Rank 8: A8, B8, B, C8, C, BC8, (BC8, C8), (C, BC8), (B8, B), (C, C8), D8, E8,
- Rank n (n>8): An, Bn, B, Cn, C, BCn, (BCn, Cn), (C, BCn), (Bn, B), (C, Cn), Dn.
Applications
showed that the affine root systems index Macdonald identitiesMacdonald identities
In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by . They include as special cases the Jacobi triple product identity, Watson's quintuple product identity, several identities found by , and a 10-fold product identity found...
used affine root systems to study p-adic algebraic groups.
- Reduced affine root systems classify affine Kac–Moody algebras, while the non-reduced affine root systems correspond to affine Lie superalgebras. showed that affine roots systems index families of Macdonald polynomials.