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 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 Macdonald (1972) and Bruhat & Tits (1972) (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 ∈ E {\displaystyle u,v\in E} {\displaystyle u,v\in E}, then it is well defined an element in V denoted as u − v {\displaystyle u-v} {\displaystyle u-v} which is the only element w such that v + w = u {\displaystyle v+w=u} {\displaystyle v+w=u}.

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

Consider the vector space F of affine-linear functions f : E ⟶ R {\displaystyle f\colon E\longrightarrow \mathbb {R} } {\displaystyle f\colon E\longrightarrow \mathbb {R} }. Having fixed a x 0 ∈ E {\displaystyle x_{0}\in E} {\displaystyle x_{0}\in E}, every element in F can be written as f ( x ) = D f ( x − x 0 ) + f ( x 0 ) {\displaystyle f(x)=Df(x-x_{0})+f(x_{0})} {\displaystyle f(x)=Df(x-x_{0})+f(x_{0})} with D f {\displaystyle Df} {\displaystyle Df} a linear function on V that doesn't depend on the choice of x 0 {\displaystyle x_{0}} {\displaystyle 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 ) = ( D f , D g ) {\displaystyle (f,g)=(Df,Dg)} {\displaystyle (f,g)=(Df,Dg)}. Set f ∨ = 2 f ( f , f ) {\displaystyle f^{\vee }={\frac {2f}{(f,f)}}} {\displaystyle f^{\vee }={\frac {2f}{(f,f)}}} and v ∨ = 2 v ( v , v ) {\displaystyle v^{\vee }={\frac {2v}{(v,v)}}} {\displaystyle v^{\vee }={\frac {2v}{(v,v)}}} for any f ∈ F {\displaystyle f\in F} {\displaystyle f\in F} and v ∈ V {\displaystyle v\in V} {\displaystyle v\in V} respectively. The identification let us define a reflection w f {\displaystyle w_{f}} {\displaystyle w_{f}} over E in the following way:

w f ( x ) = x − f ∨ ( x ) D f . {\displaystyle w_{f}(x)=x-f^{\vee }(x)Df.} {\displaystyle w_{f}(x)=x-f^{\vee }(x)Df.}

By transposition w f {\displaystyle w_{f}} {\displaystyle w_{f}} acts also on F as

w f ( g ) = g − ( f ∨ , g ) f . {\displaystyle w_{f}(g)=g-(f^{\vee },g)f.} {\displaystyle w_{f}(g)=g-(f^{\vee },g)f.}

An affine root system is a subset S ⊂ F {\displaystyle S\subset F} {\displaystyle S\subset F} such that:

  1. S spans F and its elements are non-constant.
  2. w a ( S ) = S {\displaystyle w_{a}(S)=S} {\displaystyle w_{a}(S)=S} for every a ∈ S {\displaystyle a\in S} {\displaystyle a\in S}.
  3. ( a , b ∨ ) ∈ Z {\displaystyle (a,b^{\vee })\in \mathbb {Z} } {\displaystyle (a,b^{\vee })\in \mathbb {Z} } for every a , b ∈ S {\displaystyle a,b\in S} {\displaystyle a,b\in S}.

The elements of S are called affine roots. Denote with w ( S ) {\displaystyle w(S)} {\displaystyle w(S)} the group generated by the w a {\displaystyle w_{a}} {\displaystyle w_{a}} with a ∈ S {\displaystyle a\in S} {\displaystyle a\in S}. We also ask

  1. w ( S ) {\displaystyle w(S)} {\displaystyle w(S)} as a discrete group acts properly on E.

This means that for any two compacts K , H ⊆ E {\displaystyle K,H\subseteq E} {\displaystyle K,H\subseteq E} the elements of w ( S ) {\displaystyle w(S)} {\displaystyle w(S)} such that w ( K ) ∩ H ≠ ∅ {\displaystyle w(K)\cap H\neq \varnothing } {\displaystyle w(K)\cap H\neq \varnothing } 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 systemNumber of orbitsDynkin 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, , , ...
E61
E71
E81
F42
F∨
4		2
G22
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): An, Bn, B∨
n, Cn, C∨
n, BCn, (BCn, Cn), (C∨
n, BCn), (Bn, B∨
n), (C∨
n, Cn), Dn.

Applications

References