Origins of Yangian R-Matrix Factors

Definitions and mathematical statement

Let $s\in{\mathbb{C}}$ denote the complex spectral parameter. The supplied record uses $R_{V,W}(s)$ for a Yangian $R$-matrix associated with representations $V,W$, and refers to its diagonal factor $R_0(s)$ and lower-triangular factor $R_-(s)$. These are source-dependent symbols: the record does not specify the underlying Yangian, the allowed representations, the tensor-product convention, the decomposition defining triangularity, the factorization order, or the normalization of the factors. Thus no particular factorization identity or universally quantified class of representations is asserted here. The domain on which the factors are defined, including any excluded singularities, also remains unspecified.

The extracted problem asks for the following interpretations of the factors in the intended source setting:

1. Obtain the diagonal factor $R_0(s)$ from the four-dimensional Chern--Simons theory of Costello--Witten--Yamazaki.
2. Identify the interpretation of the lower-triangular factor $R_-(s)$ in the geometric setting of Maulik--Okounkov.

The supplied record does not specify the Chern--Simons setup, the relevant geometric objects, or the precise correspondence by which either factor is to be recovered. Consequently, these are research questions with unresolved formulation data, rather than a fully specified theorem or existence conjecture.

The original passage also asks how to interpret $R_{V,W}(s)$ as a function of one complex variable and how to construct it explicitly. According to the supplied literature assessment dated 2026-08-22, the report gives explicit meromorphic constructions answering those first two questions, while the two interpretation questions above remain open. This is supplied dated curation, not an independently verified mathematical or literature conclusion.

Short English statement

Explain how the diagonal factor of the source's Yangian $R$-matrix arises from four-dimensional Chern--Simons theory, and what its lower-triangular factor means in Maulik--Okounkov geometry. The algebra, representations, factor conventions, and intended correspondences require clarification from the cited report. The supplied partial-solution label concerns the broader original passage and is not independently verified.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30004877$, title "Origins of Yangian R-Matrix Factors," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. This draft uses the supplied complete registry record.
• Supplied source citation: Mini-Workshop: Three Facets of R-Matrices (2022), Oberwolfach Reports, pp. 2791--2825, DOI 10.4171/owr/2021/51. https://doi.org/10.4171/owr/2021/51. The registry separately gives proposed year 2021 and names no proposer.
• Supplied literature source labeled "Appel--Vlaar": https://arxiv.org/abs/2007.09218. No title or specific supporting passage is supplied in the record.
• Attribution in the problem: four-dimensional Chern--Simons theory of Costello--Witten--Yamazaki and the geometric setting of Maulik--Okounkov. The record supplies no separate references for these attributions.
• Literature checked at: 2026-08-22. The supplied dated curation labels the broader problem partially solved, attributes explicit meromorphic constructions for the first two original questions to the report, and retains the Chern--Simons and Maulik--Okounkov interpretations as open questions. This assessment is not independently verified here; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a check against the official Oberwolfach report. This is supplied verification metadata, not a verification performed for this draft.