2 00 6 Finite - Dimensional Representations of Hyper Loop Algebras

We study finite-dimensional representations of hyper loop algebras, i.e., the hyperalgebras over a field of positive characteristic associated to nontwisted affine Kac-Moody algebras, or rather, to the underlying loop algebras. The main results are the classification of the irreducible modules, a version of Steinberg’s Tensor Product Theorem, and the construction of positive characteristic analogues of the Weyl modules as defined by Chari and Pressley in the characteristic zero case. Assuming a natural conjecture on a tensor product decomposition for these Weyl modules, we describe the blocks of the underlying abelian category. We also start the study of reduction modulo p and prove that every irreducible module of the hyper loop algebras can be constructed as a quotient of a module obtained by a certain reduction modulo p process applied to suitable characteristic zero modules.