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 ).

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

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

, every element in F can be written as

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

{\displaystyle (f,g)=(Df,Dg)}

The identification let us define a reflection

acts also on F as An affine root system is a subset

such that: The elements of S are called affine roots.

the group generated by the

We also ask This means that for any two compacts

are a finite number.

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.