Operator Relations on Yified Khovanov–Rozansky Homology

Definitions and mathematical statement

Let $L$ be a link and let $H_y(L)$ denote its y-ified Khovanov--Rozansky homology in the construction intended by the supplied source. The source describes operators $x_i,y_i$, a family $F_k$ for $k=1,2,\ldots$, and operators $f,h$ associated with an $\mathfrak{sl}_2$-action extending $F_2$. The meaning and range of the index $i$, the coefficient system, the gradings, and the precise homology construction are not specified in the supplied record.

For each such operator action, write \[ \mathcal A_L=\langle x_i,y_i,F_k,f,h\rangle \] for the associative algebra generated by these operators under addition, scalar multiplication in the intended coefficient system, and composition. A relation means a finite linear combination of words in these generators whose induced operator on $H_y(L)$ is zero. Once the coefficient system and generator index sets are fixed, a presentation amounts to determining the kernel of the evaluation homomorphism from the corresponding free associative algebra to $\mathcal A_L$.

Problem. Determine the full algebra of relations among $x_i,y_i,F_k,f,h$ acting on y-ified Khovanov--Rozansky homology. The source does not specify whether the desired relations are all relations for each individual link $L$, the relations holding simultaneously for every link, or a presentation of a separately defined algebra acting on every such homology. These targets need not coincide, so this distinction remains unresolved.

The supplied dated assessment lists the relations \[ [F_k,F_m]=0,\qquad [F_k,x_i]=0,\qquad [F_k,y_i]=k x_i^{k-1} \] for all applicable $i$ and positive integers $k,m$, and states that $F_2$ extends to an $\mathfrak{sl}_2$-action. Here the displayed brackets reproduce the supplied commutator notation; under the ordinary convention, $[A,B]=AB-BA$. The record does not settle whether graded commutators are intended, nor does it specify the normalization identifying $F_2,f,h$ with an $\mathfrak{sl}_2$-triple. Consequently, no additional bracket relations involving $f$ or $h$ are imposed in this draft.

As background, the extracted source states that any complex of the form $\mathrm{HH}^{i}(F_\beta)$ admits an action of a differential graded algebra denoted $\mathrm{CA}$, and that the resulting $\mathrm{CA}$-module is a link invariant. It describes passage to y-ified homology as a certain form of Koszul duality and states that the y-ified Khovanov--Rozansky homology of any link admits commuting operators $F_1,F_2,\ldots$. The definitions of $\mathrm{HH}^{i}(F_\beta)$, $\beta$, and $\mathrm{CA}$, and the relevant equivalence notion for the module invariant, are absent. The accompanying extracted formula for $F_k$ contains displaced summation and derivative symbols and an undefined term $u_k$; it does not provide a reliable operator definition. These background assertions and the listed relations are supplied source claims, not independently verified results in this draft.

Short English statement

Determine all relations among the operators $x_i,y_i,F_k,f,h$ acting on y-ified Khovanov--Rozansky homology. The supplied record lists some commutator relations and an $\mathfrak{sl}_2$-extension of $F_2$, but leaves the coefficient system, operator conventions, and intended meaning of a complete presentation unspecified. It also leaves unclear whether the target concerns each link separately or relations valid for every link.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004888$, title: Operator Relations on Yified Khovanov--Rozansky Homology; exact source problem code: catalogue entry. No individual proposer is supplied; the recorded proposed year is 2021.
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2022), Oberwolfach Reports, pp. 2827--2891, DOI 10.4171/owr/2021/52. https://doi.org/10.4171/owr/2021/52. The supplied citation uses 2022, while the DOI and registry year information refer to 2021; these details are retained without reconciliation.
• Supplied literature reference labeled Gorsky--Hogancamp--Mellit: https://arxiv.org/abs/2103.01212. The record supplies no title or pinpoint reference for the asserted operator relations.
• Supplied literature reference labeled OWR report: https://doi.org/10.4171/owr/2021/52.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not an independently verified assessment. It labels the problem partially solved, lists the displayed commutator relations and the $\mathfrak{sl}_2$-extension, and states that the complete algebra of relations remains open. The background explicitly calls this dated literature triage derived from supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked directly and the clean statement restored definitions and removed extraction artifacts or source-dependent wording. This is a supplied verification claim; no source visit or independent verification is claimed here.