Representations of Quantum Affinizations and Fusion Product

Abstract

In this paper we study general quantum affinizations $\mathcal{U}_q(\widehat{\mathfrak{g}})$ 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 $\text{Rep}(\mathcal{U}_q(\widehat{\mathfrak{g}}))$ of the integrable representations that we classified. We propose a new construction of tensor products in a larger category by using the Drinfel'd new coproduct (it cannot directly be used for $\text{Rep}(\mathcal{U}_q(\widehat{\mathfrak{g}}))$ because it involves infinite sums). In particular, we prove that * is a fusion product (a product of representations is a representation).