Unified Construction of Deformed W-Algebras

Definitions and mathematical statement

Objects and unresolved definitions. The supplied record concerns deformed $W$-algebras described as being of exceptional and higher-rank types. It does not specify the algebras or examples intended, their defining relations, deformation parameters, parameter restrictions, underlying coefficient fields, or the meaning of "higher-rank." Consequently, no particular family of algebras can be defined uniquely from the supplied evidence. The terms "structure" and "one uniform construction" are also not given mathematical criteria.

Statement of the supplied research problem. Determine the structure of the intended deformed $W$-algebras of exceptional and higher-rank types, and decide whether the examples under discussion arise from one uniform construction. The original formulation asks: "what is the nature of these algebras? Are these examples a part of one construction?"

Qualification. This is a research question whose objects and success criteria remain unspecified. A precise mathematical formulation requires identification of the examples, their deformation data and admissible parameters, the structural properties sought, and the sense in which a common construction must recover each example. The supplied record does not determine these choices, so it does not support a uniquely quantified conjecture or classification statement.

Short English statement

Describe the intended exceptional and higher-rank deformed $W$-algebras and determine whether a single construction accounts for the examples mentioned in the source discussion. The supplied record omits the examples and the meaning of a uniform construction, so the target remains underdetermined. Its open status is a supplied curation label, not an independently verified mathematical conclusion.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $30005072$, title "Unified Construction of Deformed W-Algebras," source problem code catalogue entry, in the "Oberwolfach Reports Open Problems" collection. No individual proposer is identified; the proposed year is recorded as 2022.
• Supplied report citation: Mini-Workshop: Recent Developments in Representation Theory and Mathematical Physics (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/16, pp. 817--848. https://doi.org/10.4171/owr/2022/16. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The citation's parenthetical year 2023 differs from the report designation 2022/16 and the recorded proposed year 2022; these supplied details have been retained without reconciliation.
• Supplied literature source: Feigin--Jimbo--Mukhin, https://arxiv.org/abs/2112.14631. No title, specific result, or passage identifying the intended examples is supplied.
• Supplied literature source: "Oberwolfach Report 2022/16," https://doi.org/10.4171/owr/2022/16.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open and states that the report leaves the general structure and unification of the exhibited deformed $W$-algebras as a main question, while the cited constructions provide examples rather than a classification or common construction. This is supplied dated curation derived from the Oberwolfach material, not an independently verified assessment. The background expressly calls for independent verification of time-sensitive status and cited claims.
• The registry assigns statement status corrected_verified and reports that the statement was reconstructed and checked against the official Oberwolfach report. These are supplied provenance assertions; no independent source verification is asserted here.