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 semi-simple 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 (except that both these papers accidentally omitted the Dynkin diagram ).

Definition 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.

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 (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 identities 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.

References