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 KacâÂÂMoody algebras. Possibly non-reduced affine root systems were introduced and classified by and (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 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.
The affine roots systems A<sub>1</sub> = B<sub>1</sub> = B = C<sub>1</sub> = C are the same, as are the pairs B<sub>2</sub> = C<sub>2</sub>, B = C, and A<sub>3</sub> = D<sub>3</sub>
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 ñ (with 2ñ a root) are colored green. The first Dynkin diagram in a series sometimes does not follow the same rule as the others.