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Proceedings of the London Mathematical Society Advance Access originally published online on June 21, 2007
Proceedings of the London Mathematical Society 2007 95(3):567-608; doi:10.1112/plms/pdm017
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© 2007 London Mathematical Society

Drinfeld coproduct, quantum fusion tensor category and applications

David Hernandez

Università di Roma ‘La aSapienza’
and
École Normale Supérieure Paris
45, Rue d’Ulm
75230 Paris cedex 05
France
david.hernandez{at}ens.fr
Current address: CNRS – UMR 8100: Laboratoire de Mathématiques de Versailles
45, avenue des Etats-Unis
78035 Versailles
France

Received 17 May 2005. Revision received 29 March 2006.

The class of quantum affinizations (or quantum loop algebras) includes quantum affine algebras and quantum toroidal algebras. In general, they have no Hopf algebra structure, but have a ‘coproduct’ (the Drinfeld coproduct) which does not produce tensor products of modules in the usual way because it is defined in a completion. In this paper we propose a new process to produce quantum fusion modules from it: for all quantum affinizations, we construct by deformation and renormalization a new (non-semi-simple) tensor category Mod. For quantum affine algebras this process is new and different from the usual tensor product. So far, for general quantum affinizations, for example for toroidal algebras, no process to produce fusion modules was known. We derive several applications from it: we construct the fusion of (finitely many) arbitrary l-highest weight modules, and prove that it is always cyclic. We establish exact sequences involving fusion of Kirillov–Reshetikhin modules related to new T-systems extending results for quantum affine algebras (Nakajima, Hernandez). Eventually, for a large class of quantum affinizations we prove that the subcategory of finite-length modules of Mod is stable under the new monoidal bifunctor.

2000 Mathematics Subject Classification 17B37 (primary), 20G42, 81R50 (secondary).


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