Non-minimality of certain irregular coherent preminimal affinizations

Adriano Moura and Fernanda Pereira

Vol. 297 (2018), No. 1, 147–193

DOI: 10.2140/pjm.2018.297.147

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

Milestones

Received: 15 January 2018

Revised: 26 June 2018

Accepted: 26 June 2018

Published: 7 October 2018