Affine root system
# 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 ).

## 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:

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

* $w(S)$ 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 $A_1$ = $B_1$ = $B^∨$ $_1$ = $C_1$ = $C^∨$ $_1$ are the same, as are the pairs $B_2$ = $C_2,$ $B^∨$ $_2$ = $C^∨$ $_2,$ and $A_3$ = $D_3$

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 |
| - | - | - |
| $A_n$ (n ≥ 1) | 2 if n=1, 1 if n≥2 | , , , , ... |
| $B_n$ (n ≥ 3) | 2 | , ,, ... |
| $B^∨$ $_n$ (n ≥ 3) | 2 | , ,, ... |
| $C_n$ (n ≥ 2) | 3 | , , , ... |
| $C^∨$ $_n$ (n ≥ 2) | 3 | , , , ... |
| $BC_n$ (n ≥ 1) | 2 if n=1, 3 if n ≥ 2 | , , , , ... |
| $D_n$ (n ≥ 4) | 1 | , , , ... |
| $E_6$ | 1 | |
| $E_7$ | 1 | |
| $E_8$ | 1 | |
| $F_4$ | 2 | |
| $F^∨$ $_4$ | 2 | |
| $G_2$ | 2 | |
| $G^∨$ $_2$ | 2 | |
| $(BC_n,$ $C_n)$ (n ≥ 1) | 3 if n=1, 4 if n≥2 | , , , , ... |
| $(C^∨$ $_n,$ $BC_n)$ (n ≥ 1) | 3 if n=1, 4 if n≥2 | , , , , ... |
| $(B_n,$ $B^∨$ $_n)$ (n ≥ 2) | 4 if n=2, 3 if n≥3 | , , ,, ... |
| $(C^∨$ $_n,$ $C_n)$ (n ≥ 1) | 4 if n=1, 5 if n≥2 | , , , , ... |

### Irreducible affine root systems by rank

    Rank 1: $A_1,$ $BC_1,$ $(BC_1,$ $C_1),$ $(C^∨$ $_1,$ $BC_1),$ $(C^∨$ $_1,$ $C_1).$
    Rank 2: $A_2,$ $C_2,$ $C^∨$ $_2,$ $BC_2,$ $(BC_2,$ $C_2),$ $(C^∨$ $_2,$ $BC_2),$ $(B_2,$ $B^∨$ $_2),$ $(C^∨$ $_2,$ $C_2),$ $G_2,$ $G^∨$ $_2.$
    Rank 3: $A_3,$ $B_3,$ $B^∨$ $_3,$ $C_3,$ $C^∨$ $_3,$ $BC_3,$ $(BC_3,$ $C_3),$ $(C^∨$ $_3,$ $BC_3),$ $(B_3,$ $B^∨$ $_3),$ $(C^∨$ $_3,$ $C_3).$
    Rank 4: $A_4,$ $B_4,$ $B^∨$ $_4,$ $C_4,$ $C^∨$ $_4,$ $BC_4,$ $(BC_4,$ $C_4),$ $(C^∨$ $_4,$ $BC_4),$ $(B_4,$ $B^∨$ $_4),$ $(C^∨$ $_4,$ $C_4),$ $D_4,$ $F_4,$ $F^∨$ $_4.$
    Rank 5: $A_5,$ $B_5,$ $B^∨$ $_5,$ $C_5,$ $C^∨$ $_5,$ $BC_5,$ $(BC_5,$ $C_5),$ $(C^∨$ $_5,$ $BC_5),$ $(B_5,$ $B^∨$ $_5),$ $(C^∨$ $_5,$ $C_5),$ $D_5.$
    Rank 6: $A_6,$ $B_6,$ $B^∨$ $_6,$ $C_6,$ $C^∨$ $_6,$ $BC_6,$ $(BC_6,$ $C_6),$ $(C^∨$ $_6,$ $BC_6),$ $(B_6,$ $B^∨$ $_6),$ $(C^∨$ $_6,$ $C_6),$ $D_6,$ $E_6,$
    Rank 7: $A_7,$ $B_7,$ $B^∨$ $_7,$ $C_7,$ $C^∨$ $_7,$ $BC_7,$ $(BC_7,$ $C_7),$ $(C^∨$ $_7,$ $BC_7),$ $(B_7,$ $B^∨$ $_7),$ $(C^∨$ $_7,$ $C_7),$ $D_7,$ $E_7,$
    Rank 8: $A_8,$ $B_8,$ $B^∨$ $_8,$ $C_8,$ $C^∨$ $_8,$ $BC_8,$ $(BC_8,$ $C_8),$ $(C^∨$ $_8,$ $BC_8),$ $(B_8,$ $B^∨$ $_8),$ $(C^∨$ $_8,$ $C_8),$ $D_8,$ $E_8,$
    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.
