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 formula_1, then it is well defined an element in "V" denoted as formula_2 which is the only element w such that formula_3.
Now suppose we have a scalar product formula_4 on "V".
This defines a metric on "E" as formula_5.
Consider the vector space "F" of affine-linear functions formula_6.
Having fixed a formula_7, very element in "F" can be written as formula_8 with formula_9 a linear function on "V" that doesn't depend on the choice of formula_10.
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 formula_11.
Set formula_12 and formula_13 for any formula_14 and formula_15 respectively.
The identification let us define a reflection formula_16 over "E" in the following way:
By transposition formula_16 acts also on "F" as
An "affine root system" is a subset formula_20 such that:
The elements of "S" are called "affine roots".
Denote with formula_21 the group generated by the formula_22 with formula_23.
We also ask