Representations of quantum affinizations and fusion product

In this paper we study general quantum affinizations U-q((g) over cap) of symmetrizable quantum Kac-Moody algebras and we develop their representation theory. We prove a triangular decomposition and we give a classication of (type 1) highest weight simple integrable representations analog to Drinfel'd-Chari-Presley one. A generalization of the q-characters morphism, introduced by Frenkel-Reshetikhin for quantum affine algebras, appears to be a powerful tool for this investigation. For a large class of quantum affinizations (including quantum affine algebras and quantum toroidal algebras), the combinatorics of q-characters give a ring structure * on the Grothendieck group Rep(U-q((g) over cap)) of the integrable representations that we classified. We propose a new construction of tenser products in a larger category by using the Drinfel'd new coproduct (it cannot directly be used for Rep(U-q((g) over cap)) because it involves infinite sums). In particular, we prove that * is a fusion product (a product of representations is a representation).