Affine root system

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The affine root system of type G2.

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 Dyn-node.pngDyn-4b.pngDyn-nodeg.pngDyn-4a.pngDyn-node.png).

Definition[edit]

Classification[edit]

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 Dyn-node.pngDyn-4ab.pngDyn-node.png, Dyn2-branch.pngDyn2-loop2.png, Dyn2-loop1.pngDyn2-nodes.pngDyn2-loop2.png, Dyn2-branch.pngDyn2-3s.pngDyn2-nodes.pngDyn2-loop2.png, ...
Bn (n ≥ 3) 2 Dyn-branch1.pngDyn-node.pngDyn-4b.pngDyn-node.png, Dyn-branch1.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-node.png,Dyn-branch1.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-node.png, ...
B
n
(n ≥ 3)
2 Dyn-branch1.pngDyn-node.pngDyn-4a.pngDyn-node.png, Dyn-branch1.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4a.pngDyn-node.png,Dyn-branch1.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4a.pngDyn-node.png, ...
Cn (n ≥ 2) 3 Dyn-node.pngDyn-4b.pngDyn-node.pngDyn-4a.pngDyn-node.png, Dyn-node.pngDyn-4b.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4a.pngDyn-node.png, Dyn-node.pngDyn-4b.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4a.pngDyn-node.png, ...
C
n
(n ≥ 2)
3 Dyn-node.pngDyn-4a.pngDyn-node.pngDyn-4b.pngDyn-node.png, Dyn-node.pngDyn-4a.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-node.png, Dyn-node.pngDyn-4a.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-node.png, ...
BCn (n ≥ 1) 2 if n=1, 3 if n ≥ 2 Dyn-node.pngDyn-4c.pngDyn-node.png, Dyn-node.pngDyn-4a.pngDyn-node.pngDyn-4a.pngDyn-node.png, Dyn-node.pngDyn-4a.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4a.pngDyn-node.png, Dyn-node.pngDyn-4a.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4a.pngDyn-node.png, ...
Dn (n ≥ 4) 1 Dyn-branch1.pngDyn-node.pngDyn-branch2.png, Dyn-branch1.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-branch2.png, Dyn-branch1.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-branch2.png, ...
E6 1 Dyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-branch2.pngDyn-3s.pngDyn-nodes.png
E7 1 Dyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.png
E8 1 Dyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-branch.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.pngDyn2-3.pngDyn2-node.png
F4 2 Dyn-node.pngDyn-3.pngDyn-node.pngDyn-4a.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.png
F
4
2 Dyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.png
G2 2 Dyn-node.pngDyn-6a.pngDyn-node.pngDyn-3.pngDyn-node.png
G
2
2 Dyn-node.pngDyn-6b.pngDyn-node.pngDyn-3.pngDyn-node.png
(BCn, Cn) (n ≥ 1) 3 if n=1, 4 if n≥2 Dyn-nodeg.pngDyn-4c.pngDyn-node.png, Dyn-nodeg.pngDyn-4a.pngDyn-node.pngDyn-4a.pngDyn-node.png, Dyn-nodeg.pngDyn-4a.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4a.pngDyn-node.png, Dyn-nodeg.pngDyn-4a.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4a.pngDyn-node.png, ...
(C
n
, BCn) (n ≥ 1)
3 if n=1, 4 if n≥2 Dyn-nodeg.pngDyn-4ab.pngDyn-node.png, Dyn-nodeg.pngDyn-4a.pngDyn-node.pngDyn-4b.pngDyn-node.png, Dyn-nodeg.pngDyn-4a.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-node.png, Dyn-nodeg.pngDyn-4a.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-node.png, ...
(Bn, B
n
) (n ≥ 2)
4 if n=2, 3 if n≥3 Dyn-node.pngDyn-4b.pngDyn-nodeg.pngDyn-4a.pngDyn-node.png, Dyn-branch1.pngDyn-node.pngDyn-4b.pngDyn-nodeg.png, Dyn-branch1.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-nodeg.png,Dyn-branch1.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-nodeg.png, ...
(C
n
, Cn) (n ≥ 1)
4 if n=1, 5 if n≥2 Dyn-nodeg.pngDyn-4ab.pngDyn-nodeg.png, Dyn-nodeg.pngDyn-4a.pngDyn-node.pngDyn-4b.pngDyn-nodeg.png, Dyn-nodeg.pngDyn-4a.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-nodeg.png, Dyn-nodeg.pngDyn-4a.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-4b.pngDyn-nodeg.png, ...

Irreducible affine root systems by rank[edit]

Rank 1: A1, BC1, (BC1, C1), (C
1
, BC1), (C
1
, C1).
Rank 2: A2, C2, C
2
, BC2, (BC2, C2), (C
2
, BC2), (B2, B
2
), (C
2
, C2), G2, G
2
.
Rank 3: A3, B3, B
3
, C3, C
3
, BC3, (BC3, C3), (C
3
, BC3), (B3, B
3
), (C
3
, C3).
Rank 4: A4, B4, B
4
, C4, C
4
, BC4, (BC4, C4), (C
4
, BC4), (B4, B
4
), (C
4
, C4), D4, F4, F
4
.
Rank 5: A5, B5, B
5
, C5, C
5
, BC5, (BC5, C5), (C
5
, BC5), (B5, B
5
), (C
5
, C5), D5.
Rank 6: A6, B6, B
6
, C6, C
6
, BC6, (BC6, C6), (C
6
, BC6), (B6, B
6
), (C
6
, C6), D6, E6,
Rank 7: A7, B7, B
7
, C7, C
7
, BC7, (BC7, C7), (C
7
, BC7), (B7, B
7
), (C
7
, C7), D7, E7,
Rank 8: A8, B8, B
8
, C8, C
8
, BC8, (BC8, C8), (C
8
, BC8), (B8, B
8
), (C
8
, C8), D8, E8,
Rank n (n>8): An, Bn, B
n
, Cn, C
n
, BCn, (BCn, Cn), (C
n
, BCn), (Bn, B
n
), (C
n
, Cn), Dn.

Applications[edit]

References[edit]