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 ).
Let E be an affine space and V the vector space of its translations. Recall that V acts faithfully and transitively on E. In particular, if , then it is well defined an element in V denoted as which is the only element w such that .
Now suppose we have a scalar product on V. This defines a metric on E as .
Consider the vector space F of affine-linear functions . Having fixed a , every element in F can be written as with a linear function on V that doesn't depend on the choice of .
Now the dual of V can be identified with V thanks to the chosen scalar product and we can define a product on F as . Set and for any and respectively. The identification let us define a reflection over E in the following way:
By transposition acts also on F as
An affine root system is a subset such that:
The elements of S are called affine roots. Denote with the group generated by the with . We also ask
This means that for any two compacts the elements of such that are a finite number.
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 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |