Affine root system

Image:G2 [affine chamber.svg|thumb|340px|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 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 .

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, then it is well defined an element in V denoted as which is the only element w such that.
Now suppose we have a scalar product on V.
This defines a metric on E as.
Consider the vector space F of affine-linear functions.
Having fixed a, every element in F can be written as with a linear function on V that doesn't depend on the choice of.
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.
Set and for any and respectively.
The identification let us define a reflection over E in the following way:
By transposition acts also on F as
An affine root system is a subset such that:
The elements of S are called affine roots.
Denote with the group generated by the with.
We also ask
This means that for any two compacts the elements of such that are a finite number.

Classification

The affine roots systems A₁ = B₁ = B = C₁ = C are the same, as are the pairs B₂ = C₂, B = C, 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 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	2 if n=1, 1 if n≥2	,,,,...
B_n	2	,,,...
B	2	,,, ...
C_n	3	,,,...
C	3	,,,...
BC_n	2 if n=1, 3 if n ≥ 2	,,,,...
D_n	1	,,,...
E₆	1
E₇	1
E₈	1
F₄	2
F	2
G₂	2
G	2
	3 if n=1, 4 if n≥2	,,,,...
	3 if n=1, 4 if n≥2	,,,,...
	4 if n=2, 3 if n≥3	,,,,...
	4 if n=1, 5 if n≥2	,,,,...

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.