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 , 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 (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, BC1), (C, C1). Rank 2: A2, C2, C, BC2, (BC2, C2), (C, BC2), (B2, B), (C, C2), G2, G. Rank 3: A3, B3, B, C3, C,