Vol. 297, No. 1, 2018

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Non-minimality of certain irregular coherent preminimal affinizations

Adriano Moura and Fernanda Pereira

Vol. 297 (2018), No. 1, 147–193

Let g be a finite-dimensional simple Lie algebra of type D or E and λ be a dominant integral weight whose support bounds the subdiagram of type D4. We study certain quantum affinizations of the simple g-module of highest weight λ which we term preminimal affinizations of order 2 (this is the maximal order for such λ). This class can be split in two: the coherent and the incoherent affinizations. If λ is regular, Chari and Pressley proved that the associated minimal affinizations belong to one of the three equivalent classes of coherent preminimal affinizations. In this paper we show that, if λ is irregular, the coherent preminimal affinizations are not minimal under certain hypotheses. Since these hypotheses are always satisfied if g is of type D4, this completes the classification of minimal affinizations for type D4 by giving a negative answer to a conjecture of Chari and Pressley stating that the coherent and the incoherent affinizations were equivalent in type D4 (this corrects the opposite claim made by the first author in a previous publication).

minimal affinizations, quantum affine algebras
Mathematical Subject Classification 2010
Primary: 17B10, 17B37
Secondary: 20G42
Received: 15 January 2018
Revised: 26 June 2018
Accepted: 26 June 2018
Published: 7 October 2018
Adriano Moura
Departamento de Matemática
Universidade Estadual de Campinas
Fernanda Pereira
Departamento de Matemática, Divisão de Ciências Fundamentais
Instituto Tecnológico de Aeronáutica
São José dos Campos