Representations of quantum affinizations and fusion product. (English) Zbl 1102.17009

Hernandez, David

Transform. Groups 10, No. 2, 163-200 (2005).

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.

Reviewer: Volodymyr Mazorchuk (Uppsala)

MSC:

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

Keywords:

quantum algebra; Kac-Moody algebra; fusion product; character; integrable representation