Cluster Monomials and Simple Quantum Affine Modules

Definitions and mathematical statement

Let $\mathfrak g$ be a simple Lie algebra of type $A$, $D$, or $E$. Write $\mathcal C_\ell$ for the monoidal subcategory of finite-dimensional quantum-affine $\mathfrak g$-modules intended by the source, and write $R_\ell=K_0(\mathcal C_\ell)$ for its Grothendieck ring. Here $[M]$ denotes the class of an object $M$, the additive relations are those arising from short exact sequences, and multiplication is induced by tensor product: $[M][N]=[M\otimes N]$.

For a specified cluster algebra structure on $R_\ell$, let $\operatorname{CM}(R_\ell)$ denote its cluster monomials: products of nonnegative integral powers of variables belonging to a single cluster, subject to the chosen convention concerning frozen variables. A nonzero module is simple if it has no nonzero proper submodule. A simple object $S$ is called real if $S\otimes S$ is simple.

The source asks whether, for every permitted $\mathfrak g$ and level $\ell$, both assertions hold: \[ \forall m\in\operatorname{CM}(R_\ell),\quad \exists S\in\mathcal C_\ell\text{ simple such that }[S]=m, \] \[ \forall S\in\mathcal C_\ell\text{ simple},\quad [S]\in\operatorname{CM}(R_\ell) \quad\Longleftrightarrow\quad S\otimes S\text{ is simple}. \] The supplied record does not specify the ground field, quantum parameter, precise quantum-affine algebra and module convention, allowed levels, defining restrictions of $\mathcal C_\ell$, or cluster algebra structure and coefficient convention on $R_\ell$. Consequently these formulas express the requested assertions conditional on those source-dependent choices; they do not supply a complete independent specification of the objects being quantified over.

Short English statement

The question asks whether every cluster monomial is the class of a simple quantum-affine module in the intended category, and whether a simple module has a cluster-monomial class exactly when its tensor square is simple. The category, level range, and cluster structure require clarification from the cited sources. The supplied status is partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry: supplied complete source record for permanent ID $30001217$, titled "Cluster Monomials and Simple Quantum Affine Modules," in the Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. The recorded proposal year is 2009; no individual proposer is supplied.
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2009), Oberwolfach Reports, pp. 801--866, DOI 10.4171/owr/2009/15. https://doi.org/10.4171/owr/2009/15. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature reference, labeled "Hernandez--Leclerc primary paper": https://arxiv.org/abs/0903.1452.
• Supplied literature reference, labeled "Hernandez--Leclerc current survey": https://arxiv.org/abs/1902.01432.
• Supplied literature reference, labeled "original OWR report": https://doi.org/10.4171/owr/2009/15.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports that the cluster-monomial/simple-object assertion is proved in important finite-type and selected quantum-affine cases, while primary surveys continue to describe the arbitrary-$\mathfrak g$, arbitrary-level version as conjectural. This is supplied dated curation derived from the OWR record, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, explaining that the complete statement was preserved while malformed or undelimited notation was normalized. This supplied editorial label does not establish mathematical correctness or resolution.