Classification of Generalized Quaternion-Type Algebras

Definitions and mathematical statement

The supplied record names the following statement the Galactic Conjecture. Its specialized classes are not defined in the record, so the formulation below is conditional on recovering the intended definitions and ambient algebraic setting from the cited report.

Let $\mathcal G$ denote the source's class of algebras of generalized quaternion type, let $\mathcal W$ denote its class of weighted surface algebras, and let $X_1,\ldots,X_9$ stand for the nine exotic algebras referred to by the source. These symbols are placeholders for the source's intended objects, not definitions or identifications of those objects. Write $A\sim_{\mathrm M}B$ for Morita equivalence: equivalence of the categories of modules over $A$ and $B$, with the module and scalar conventions to be recovered from the source.

The supplied conjecture asks whether \[ \forall A\in\mathcal G,\qquad \left(\exists B\in\mathcal W:\ A\sim_{\mathrm M}B\right) \ \lor\ \left(\exists i\in\{1,\ldots,9\}:\ A\sim_{\mathrm M}X_i\right). \] The record does not specify the base field or ring, characteristic restrictions, the defining conditions for generalized quaternion type or weighted surface algebras, or the identities and parameter ranges of the nine exotic algebras. Consequently, this is a faithful schematic statement of the supplied target; a self-contained precise classification conjecture requires those missing definitions and hypotheses.

Short English statement

The Galactic Conjecture asks whether every algebra of generalized quaternion type is Morita equivalent to a weighted surface algebra or one of nine specified exotic algebras. The supplied record does not define these classes or identify the nine exceptions, so the intended algebraic setting and classification targets require human review.

Sources

• Local UnsolvedMath registry record, permanent ID $30004371$, title Classification of Generalized Quaternion-Type Algebras, exact source problem code catalogue entry, in the supplied Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2020; no proposer is identified.
• Supplied source citation: Representation Theory of Quivers and Finite Dimensional Algebras (2021), Oberwolfach Reports, pp. 143--230, DOI 10.4171/owr/2020/3. Supplied source URL and literature-source link, labeled "Oberwolfach report": https://doi.org/10.4171/owr/2020/3. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• The registry labels the problem open and the statement corrected_verified. Its supplied verification says that the statement was checked against the official report and preserves the report's hypotheses, notation, and open mathematical target. These are supplied registry assertions, not independent verification in this draft.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open but says that the extracted fragment contains only a conjecture lead-in and omits the mathematical claim, so resolution cannot be established from that record. This is supplied dated curation, not independently verified. It conflicts with the current record's explicit conjecture and its supplied statement-verification assertion. The background requests independent verification of time-sensitive status and cited claims; no updated literature assessment is asserted here.