When do cluster algebras equal their upper cluster algebras?

Determine general criteria under which a (quantum) cluster algebra A equals its upper cluster algebra U, and under which the partially compactified cluster algebra \overline{A} equals the partially compactified upper cluster algebra \overline{U}, across broad classes of seeds and ranks. This includes identifying structural or combinatorial conditions on the exchange data (e.g., rank, full-rank property, optimization of frozen variables) that guarantee the equalities A = U and \overline{A} = \overline{U}.

Background

The paper extends based cluster algebras from finite ranks to infinite ranks and proves A = U for certain infinite rank (quantum) cluster algebras associated with double Bott-Samelson cells and related constructions. Although these results provide new cases where the equality holds, the authors emphasize that, in general, characterizing when A equals U (and similarly for partial compactifications) remains largely unresolved.

The remark situates the problem within existing literature, noting a list of known cases and recent developments, and frames the paper’s contribution as progress within the infinite rank setting. A comprehensive solution would unify criteria across different families of seeds and exchange matrices, potentially leveraging optimization of frozen variables, injective-reachability, and other structural properties.