# 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.