Non-Minimality of Certain Irregular Coherent Preminimal Affinizations

TL;DR

This paper investigates the minimality of certain quantum affinizations of simple Lie algebra modules, revealing that coherent preminimal affinizations are not always minimal in irregular cases, especially for type D4.

Contribution

It demonstrates that coherent preminimal affinizations are not minimal in irregular cases, completing the classification for type D4 and correcting previous conjectures.

Findings

Coherent preminimal affinizations are not minimal for irregular weights.

The classification of minimal affinizations for type D4 is completed.

Counterexamples to the conjecture of equivalence between coherent and incoherent affinizations are provided.

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 two (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-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).