Self-Extensions of Simple Quantum-Affine Modules

Definitions and mathematical statement

The supplied record does not specify the ground field, the quantum-affine algebra, its deformation parameter, or the category in which extensions are computed. Consequently, the following is a conditional formulation of the recorded characterization problem, pending those choices.

Let $k$ denote the intended ground field, let $A$ denote the intended quantum-affine algebra over $k$, and let $\mathcal C$ denote the intended $k$-linear abelian category of $A$-modules. For objects $V$ of $\mathcal C$, define $\operatorname{Ext}^{1}_{\mathcal C}(V,V)$ to be the $k$-vector space of Yoneda equivalence classes of short exact sequences in $\mathcal C$, \[ 0\longrightarrow V\longrightarrow E\longrightarrow V\longrightarrow 0, \] where equivalence means an isomorphism of short exact sequences inducing the identity on both end terms, and addition is the Baer sum. A module is simple if it is nonzero and has no nonzero proper submodule.

For the intended fixed choices of $k$, $A$, and $\mathcal C$, characterize, up to $A$-module isomorphism, all simple modules $V\in\mathcal C$ such that \[ \dim_k V<\infty \qquad\text{and}\qquad \dim_k\operatorname{Ext}^{1}_{\mathcal C}(V,V)=1. \] The requested characterization should give necessary and sufficient conditions for this equality for every simple finite-dimensional module in the intended category. The unspecified choices above are unresolved qualifications, not an assertion that the same characterization applies to every quantum-affine algebra or every extension category. In particular, the supplied record does not state a precise equivalence between this condition and primeness.

Short English statement

Determine exactly which simple finite-dimensional quantum-affine modules have a one-dimensional space of self-extensions. A precise classification requires specifying the ground field, quantum-affine algebra, deformation parameter, and extension category. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002004$, title: Self-Extensions of Simple Quantum-Affine Modules; source problem code catalogue entry. The record assigns the year 2012 and names no individual proposer.
• Supplied report provenance: Enveloping Algebras and Geometric Representation Theory (2012), Oberwolfach Reports, pp. 733--809, DOI 10.4171/owr/2012/13. https://doi.org/10.4171/owr/2012/13. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; no precise problem page or passage is supplied.
• Supplied literature reference: Prime Representations from a Homological Perspective. https://arxiv.org/abs/1112.6376.
• Supplied literature reference: Prime Representations from a Homological Perspective. https://doi.org/10.1007/s00209-012-1088-7.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the self-extension criterion is proved in type $\mathfrak{sl}_2$ and for broad classes of prime representations, while a converse characterization for all simple quantum-affine representations remains conjectural. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.