Affine root system

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) (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 [math]\displaystyle{ u,v \in E }[/math], then it is well defined an element in V denoted as [math]\displaystyle{ u-v }[/math] which is the only element w such that [math]\displaystyle{ v+w=u }[/math].

Now suppose we have a scalar product [math]\displaystyle{ (\cdot,\cdot) }[/math] on V. This defines a metric on E as [math]\displaystyle{ d(u,v)=\vert(u-v,u-v)\vert }[/math].

Consider the vector space F of affine-linear functions [math]\displaystyle{ f\colon E\longrightarrow \mathbb{R} }[/math]. Having fixed a [math]\displaystyle{ x_0\in E }[/math], every element in F can be written as [math]\displaystyle{ f(x)=Df(x-x_0)+f(x_0) }[/math] with [math]\displaystyle{ Df }[/math] a linear function on V that doesn't depend on the choice of [math]\displaystyle{ x_0 }[/math].

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 [math]\displaystyle{ (f,g)=(Df,Dg) }[/math]. Set [math]\displaystyle{ f^\vee =\frac{2f}{(f,f)} }[/math] and [math]\displaystyle{ v^\vee =\frac{2v}{(v,v)} }[/math] for any [math]\displaystyle{ f\in F }[/math] and [math]\displaystyle{ v\in V }[/math] respectively. The identification let us define a reflection [math]\displaystyle{ w_f }[/math] over E in the following way:

[math]\displaystyle{ w_f(x)=x-f^\vee(x)Df }[/math]

By transposition [math]\displaystyle{ w_f }[/math] acts also on F as

[math]\displaystyle{ w_f(g)=g-(f^\vee,g)f }[/math]

An affine root system is a subset [math]\displaystyle{ S\in F }[/math] such that:

1. S spans F and its elements are non-constant.
2. [math]\displaystyle{ w_a(S)=S }[/math] for every [math]\displaystyle{ a\in S }[/math].
3. [math]\displaystyle{ (a,b^\vee)\in\mathbb{Z} }[/math] for every [math]\displaystyle{ a,b\in S }[/math].

The elements of S are called affine roots. Denote with [math]\displaystyle{ w(S) }[/math] the group generated by the [math]\displaystyle{ w_a }[/math] with [math]\displaystyle{ a\in S }[/math]. We also ask

4. [math]\displaystyle{ w(S) }[/math] as a discrete group acts properly on E.

This means that for any two compacts [math]\displaystyle{ K,H\subseteq E }[/math] the elements of [math]\displaystyle{ w(S) }[/math] such that [math]\displaystyle{ w(K)\cap H\neq \varnothing }[/math] are a finite number.

Classification

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	, , , , ...

Irreducible affine root systems by rank

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): A_n, B_n, B∨n, C_n, C∨n, BC_n, (BC_n, C_n), (C∨n, BC_n), (B_n, B∨n), (C∨n, C_n), D_n.

Applications

• (Macdonald 1972) showed that the affine root systems index Macdonald identities
• (Bruhat Tits) 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.
• (Macdonald 2003) showed that affine roots systems index families of Macdonald polynomials.

References

• Bruhat, F.; Tits, Jacques (1972), "Groupes réductifs sur un corps local", Publications Mathématiques de l'IHÉS 41: 5–251, doi:10.1007/bf02715544, ISSN 1618-1913, http://www.numdam.org/item?id=PMIHES_1972__41__5_0 
• Macdonald, I. G. (1972), "Affine root systems and Dedekind's η-function", Inventiones Mathematicae 15 (2): 91–143, doi:10.1007/BF01418931, ISSN 0020-9910, Bibcode: 1971InMat..15...91M 
• Macdonald, I. G. (2003), Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, 157, Cambridge: Cambridge University Press, pp. x+175, ISBN 978-0-521-82472-9