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Representations of quantum affinizations and fusion product. (English) Zbl 1102.17009
The quantum affine algebras have two different realizations: the usual Drinfeld-Jimbo realization and an alternative realization via quantum affinization of quantum algebras of finite type. The quantum affinization process extends to all symmetrizable Kac-Moody algebras giving a new class of algebras, called quantum affinizations. In the paper under review the author studies general quantum affinizations and develops their representation theory. He constructs a triangular decomposition for such algebras and classifies the type 1 simple highest weight integrable representations.
The central notion in the author’s study is that of a $$q$$-character, defined representation-theoretically. The author shows that such character can be given a nice combinatorial description. Furthermore, the combinatorics of $$q$$-characters gives a ring structure on the Grothendieck group of certain integrable representations. Finally, it is shown that this ring structure corresponds to a certain fusion product.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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