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 and  (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: 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 1. $w(S)$ 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.

Irreducible affine root systems by rank

 * Rank 1: A1, BC1, (BC1, C1), (C$∨ n$, BC1), (C$∨ n$, C1).
 * Rank 2: A2, C2, C$∨ 4$, BC2, (BC2, C2),  (C$∨ 2$, BC2), (B2, B$∨ n$), (C$∨ n$, C2),  G2, G$∨ n$.
 * Rank 3: A3, B3, B$∨ 1$, C3, C$∨ 1$, BC3, (BC3, C3), (C$∨ 2$, BC3), (B3, B$∨ 2$), (C$∨ 2$, C3).
 * Rank 4: A4, B4, B$∨ 2$, C4, C$∨ 2$, BC4, (BC4, C4), (C$∨ 3$, BC4), (B4, B$∨ 3$), (C$∨ 3$, C4), D4, F4, F$∨ 3$.
 * Rank 5: A5, B5, B$∨ 3$, C5, C$∨ 4$, BC5, (BC5, C5), (C$∨ 4$, BC5), (B5, B$∨ 4$), (C$∨ 4$, C5), D5.
 * Rank 6: A6, B6, B$∨ 4$, C6, C$∨ 4$, BC6, (BC6, C6), (C$∨ 5$, BC6), (B6, B$∨ 5$), (C$∨ 5$, C6), D6, E6,
 * Rank 7: A7, B7, B$∨ 5$, C7, C$∨ 5$, BC7, (BC7, C7), (C$∨ 6$, BC7), (B7, B$∨ 6$), (C$∨ 6$, C7), D7, E7,
 * Rank 8: A8, B8, B$∨ 6$, C8, C$∨ 6$, BC8, (BC8, C8), (C$∨ 7$, BC8), (B8, B$∨ 7$), (C$∨ 7$, C8), D8, E8,
 * Rank n (n>8): An, Bn, B$∨ 7$, Cn, C$∨ 7$, BCn, (BCn, Cn), (C$∨ 8$, BCn), (Bn, B$∨ 8$), (C$∨ 8$, Cn), Dn.

Applications

 * showed that the affine root systems index Macdonald identities
 * 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.
 * showed that affine roots systems index families of Macdonald polynomials.