Representations of Quantum Affinizations and Fusion Product

2003/12/31 by David Hernandez
Mathematics · #math.QA #math.RT

paper · pdf

published as Transformation Groups 10 (2005), no. 2, 163--200 · 33 pages, 1 figure. Accepted for publication in Transformation Groups

arxiv created 2004/09/02 · arxiv updated 2009/12/01

Abstract

In this paper we study general quantum affinizations \U_q(\Glie) 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-Pressley 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(\Glie)) 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 can not directly be used for Rep(\U_q(\Glie)) because it involves infinite sums). In particular we prove that * is a fusion product (a product of representations is a representation).

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