Composition Algebras of Affine Type

The aim of this paper is to study the structure of the composition algebras of affine type. It turns out that they have a triangular decomposition corresponding to the division of the indecomposables into the preprojectives, the regulars, and the preinjectives. By the recent Ringel–Green theorem the composition algebra can be twisted in order to obtain the positive partU + of the Drinfeld–Jimbo quantized enveloping algebraU = U − U0 U + of the corresponding Kac–Moody algebra, and one obtains a corresponding triangular decomposition forU + , in particular, a natural basis ofU + in terms ofA-representations.