Affine root system explained

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 (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

u,v \in E , then it is well defined an element in V denoted as u - v which is the only element w such that v + w = u .

Now suppose we have a scalar product

( ⋅ , ⋅ ) on V.This defines a metric on E as d(u,v) = \vert(u - v,u - v)\vert .f\colon E\longrightarrow R .Having fixed a x _{0\in} E , every element in F can be written as f(x) = Df(x - x _{0)+f(x} _{0)} with Df a linear function on V that doesn't depend on the choice of x _{0} .

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

(f,g) = (Df,Dg) .Set f ^{\vee} = (2 f)/((f,f)) and v ^{\vee} = (2 v)/((v,v)) for any f\in F and v\in V respectively.The identification let us define a reflection w _{f} over E in the following way:w ^{\vee(x)Df} _{f(x) = x - f} By transposition w _{f} acts also on F asw ^{\vee,g)f} _{f(g) = g - (f}

An affine root system is a subset

S\in F such that:The elements of S are called affine roots.Denote with w(S) the group generated by the w _{a} with a\in S .We also askThis means that for any two compacts K,H\subseteq E the elements of w(S) such that w(K)\cap H ≠ \varnothing 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): A_n, B_n, B, C_n, C, BC_n, (BC_n, C_n), (C, BC_n), (B_n, B), (C, C_n), D_n.

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.