Affine root system

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 ).

The affine root system of type G₂.

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 $(\cdot ,\cdot )$ on V. This defines a metric on E as $d(u,v)=\vert (u-v,u-v)\vert$.

Consider the vector space F of affine-linear functions $f\colon E\longrightarrow \mathbb {R}$. 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 $f^{\vee }={\frac {2f}{(f,f)}}$ and $v^{\vee }={\frac {2v}{(v,v)}}$ 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:

$w_{f}(x)=x-f^{\vee }(x)Df$

By transposition $w_{f}$ acts also on F as

$w_{f}(g)=g-(f^{\vee },g)f$

An affine root system is a subset $S\in F$ such that:

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

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 ask

4. $w(S)$ as a discrete group acts properly on E.

This means that for any two compacts $K,H\subseteq E$ the elements of $w(S)$ such that $w(K)\cap H\neq \varnothing$ 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 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 (1972) 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, MR 0327923, S2CID 125864274
• Macdonald, I. G. (1972), "Affine root systems and Dedekind's η-function", Inventiones Mathematicae, 15 (2): 91–143, Bibcode:1971InMat..15...91M, doi:10.1007/BF01418931, ISSN 0020-9910, MR 0357528, S2CID 122115111
• Macdonald, I. G. (2003), Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge: Cambridge University Press, pp. x+175, ISBN 978-0-521-82472-9, MR 1976581