Vol. 297, No. 1, 2018

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Non-minimality of certain irregular coherent preminimal affinizations

Vol. 297 (2018), No. 1, 147–193
Abstract

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of type $D$ or $E$ and $\lambda$ be a dominant integral weight whose support bounds the subdiagram of type ${D}_{4}$. We study certain quantum affinizations of the simple $\mathfrak{g}$-module of highest weight $\lambda$ which we term preminimal affinizations of order 2 (this is the maximal order for such $\lambda$). This class can be split in two: the coherent and the incoherent affinizations. If $\lambda$ 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 $\lambda$ is irregular, the coherent preminimal affinizations are not minimal under certain hypotheses. Since these hypotheses are always satisfied if $\mathfrak{g}$ is of type ${D}_{4}$, this completes the classification of minimal affinizations for type ${D}_{4}$ by giving a negative answer to a conjecture of Chari and Pressley stating that the coherent and the incoherent affinizations were equivalent in type ${D}_{4}$ (this corrects the opposite claim made by the first author in a previous publication).

Keywords
minimal affinizations, quantum affine algebras
Mathematical Subject Classification 2010
Primary: 17B10, 17B37
Secondary: 20G42