Welding Set-Theoretic Yang–Baxter Solutions

Definitions and mathematical statement

Let $X$ be a set. Use the braid-form convention: a set-theoretic Yang--Baxter solution is a bijection $r\colon X^2\to X^2$ satisfying \[ (r\times\mathrm{id}_X)(\mathrm{id}_X\times r)(r\times\mathrm{id}_X) = (\mathrm{id}_X\times r)(r\times\mathrm{id}_X)(\mathrm{id}_X\times r) \quad\text{on }X^3. \] Bijectivity is required for the stated braid-group actions; no additional nondegeneracy assumption is supplied. For $n\geq2$ and $1\leq i<n$, let \[ r_i=\mathrm{id}_X^{\times(i-1)}\times r\times\mathrm{id}_X^{\times(n-i-1)}. \] These maps define $\rho_{r,n}\colon B_n\to\operatorname{Sym}(X^n)$ by $\rho_{r,n}(\sigma_i)=r_i$, where $B_n$ is the braid group with its usual generators $\sigma_i$.

To specify the extension problem precisely, let $F_n=\langle g_1,\ldots,g_n\rangle$ and use the algebraic realization supplied for the welded braid group: \[ WB_n=\left\{\phi\in\operatorname{Aut}(F_n): \begin{array}{l} \text{there exist }\pi\in S_n\text{ and }w_1,\ldots,w_n\in F_n\\ \text{such that }\phi(g_j)=w_jg_{\pi(j)}w_j^{-1}\text{ for every }j \end{array}\right\}. \] Fix the Artin inclusion $\iota_n\colon B_n\hookrightarrow WB_n$ by $\iota_n(\sigma_i)=A_i$, where \[ A_i(g_i)=g_ig_{i+1}g_i^{-1},\qquad A_i(g_{i+1})=g_i,\qquad A_i(g_j)=g_j\quad(j\notin\{i,i+1\}). \] The broad question in the supplied clean statement is to characterize those $(X,r)$ for which \[ \forall n\geq2\quad\exists\ \widetilde\rho_n\colon WB_n\longrightarrow\operatorname{Sym}(X^n) \quad\text{a group homomorphism such that}\quad \widetilde\rho_n\circ\iota_n=\rho_{r,n}. \]

The accompanying exposition focuses on extensions determined by one additional local map. To state this version without inventing the omitted compatibility equations, let $P_i\in WB_n$ interchange $g_i$ and $g_{i+1}$ and fix the other free generators. Ask for a single bijection $s\colon X^2\to X^2$, satisfying $s^2=\mathrm{id}_{X^2}$ and the same braid-form Yang--Baxter equation, such that for every $n\geq2$ there is a homomorphism $\widetilde\rho_n$ as above with \[ \widetilde\rho_n(P_i)=s_i, \qquad s_i=\mathrm{id}_X^{\times(i-1)}\times s\times\mathrm{id}_X^{\times(n-i-1)} \quad(1\leq i<n). \] This gives a precise uniform local interpretation of the source's welded-pair compatibility requirement. The supplied record does not establish that arbitrary extensions in the broad question must have this form.

Supplied qualifications and unresolved target. The original excerpt reports that a representation associated with a welded pair $(X,r,s)$ is isomorphic to one associated with a derived pair $(X,r^{(s)},t)$, where $t(a,b)=(b,a)$. It describes this as answering an earlier open question. This is a reported source claim, not a theorem verified here: the construction of $r^{(s)}$, its full hypotheses, and the precise isomorphism statement are absent. The dated curation labels the record partially solved and identifies weighted $2$-cocycle and further welded-pair classification questions as remaining open. However, the original excerpt ends at "Question 1. Does using general second components s become relevant when one enriches the induced representations with w". Thus neither the weighted construction nor the remaining classification target can be stated precisely from the supplied evidence.

Short English statement

Characterize the bijective set-theoretic Yang--Baxter solutions whose actions on every $X^n$ extend from $B_n$ to $WB_n$. The source particularly considers extensions using a common involutive Yang--Baxter solution $s$ for the permutation generators. It reports a reduction to the twist solution for ordinary representations, while the supplied dated curation describes weighted $2$-cocycle and further classification questions as open. The excerpt truncates before those questions are specified, so their precise statements remain unresolved.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30006491$, exact source problem code catalogue entry, title Welding Set-Theoretic Yang--Baxter Solutions, collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2025.
• Supplied report citation: Mini-Workshop: The Yang--Baxter Equation and Representations of Braid Groups (2026), Oberwolfach Reports, pp. 2633--2672, DOI 10.4171/owr/2025/49. https://doi.org/10.4171/owr/2025/49. The literature-source label is Oberwolfach Report 49/2025. The supplied workshop year and report-year designation differ and have not been reconciled.
• Supplied literature reference: Yang--Baxter 2-cocycle invariants. https://arxiv.org/abs/1707.02250. No authors or specific result supporting the weighted extension question are identified in the supplied record.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the record partially_solved, reports that the basic welded-pair representation question is answered by an explicit isomorphism to a derived pair with the twist solution, and describes weighted $2$-cocycle and further welded-pair classification questions in the same record as open. The record cautions that time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that the official report was checked and its notation repaired. These are supplied provenance claims, not verification performed for this draft. The excerpt's references [1], [2], and [3] have no accompanying bibliography: [1] concerns the earlier open question, [2] the antecedent welded-pair idea, and [3] further details about welded braid groups. Their identities cannot be recovered from the supplied text.