Prime Quotients of Yang–Baxter Structure Algebras

Definitions and mathematical statement

The supplied record leaves the admissible class of coefficients and solutions unspecified. The following notation uses the usual structure-monoid convention conditionally; it does not resolve those missing hypotheses.

Let $X$ be a set and $r\colon X\times X\to X\times X$ a map. In the braid convention, the set-theoretic Yang--Baxter equation is \[ (r\times\mathrm{id}_X)(\mathrm{id}_X\times r)(r\times\mathrm{id}_X) = (\mathrm{id}_X\times r)(r\times\mathrm{id}_X)(\mathrm{id}_X\times r) \quad\text{on }X^3. \] Whether the source additionally requires bijectivity, involutivity, nondegeneracy, or finiteness of $X$ is not specified. Under the standard presentation convention, define the structure monoid by \[ M(X,r)=\left\langle X\ \middle|\ xy=uv\text{ whenever }r(x,y)=(u,v)\right\rangle. \] Write $A=kM(X,r)$ for its monoid algebra: its elements are finite formal sums $\sum_{m\in M(X,r)}a_m m$, with coefficients in $k$, and multiplication is induced by monoid multiplication and distributivity. The record does not state the hypotheses on $k$, including whether it must be a field or whether its characteristic is restricted.

A proper two-sided ideal $P\subsetneq A$ is prime if, for all two-sided ideals $I,J$ of $A$, \[ IJ\subseteq P\ \Longrightarrow\ I\subseteq P\text{ or }J\subseteq P. \] For each such $P$, put $B_P=A/P$.

Problem. For every admissible choice of $k$, $(X,r)$, and prime ideal $P$ in the intended source setting, determine the structure of $B_P$. In particular, characterize when $B_P$ is a domain, a maximal order, Koszul, Auslander-regular, or Cohen--Macaulay, and determine any further homological properties intended by the source. Here the domain question asks whether \[ \forall a,b\in B_P,\qquad ab=0\ \Longrightarrow\ a=0\text{ or }b=0. \] This condition is stronger than the primeness already ensured by the choice of $P$.

Determine also when $B_P$ arises from a related Yang--Baxter solution and what implications the structure and properties of $B_P$ have for the original solution $(X,r)$. The source does not specify the permitted relation between solutions, the algebraic construction meant by "arises from," or the conclusions sought about $(X,r)$; these parts therefore remain characterization requests with unresolved scope.

The property requests also require conventions absent from the record. The displayed monoid relations give a natural length grading on $A$, but an arbitrary prime ideal $P$ need not be homogeneous, so an inherited grading for the usual Koszul question is not assured. The ambient quotient ring and order convention for "maximal order," the sidedness conventions for Auslander regularity, and the dimension function and module class for the Cohen--Macaulay condition are unspecified. No finiteness, Noetherian, grading, or homological hypothesis needed for those interpretations is asserted here. The admissible input class and these conventions require human mathematical review before this can become a fully specified characterization problem.

Short English statement

For each prime ideal of a Yang--Baxter structure algebra, characterize its quotient, especially its domain, order, graded, and homological properties. Determine when the quotient is associated with a related solution and what it reveals about the original solution. The coefficient and solution hypotheses, several property conventions, and the meaning of a related solution remain unspecified. The supplied partially solved status is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004304$, exact source problem code catalogue entry, title: Prime Quotients of Yang--Baxter Structure Algebras. The record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied original report citation: Mini-Workshop: Algebraic Tools for Solving the Yang--Baxter Equation (2020), Oberwolfach Reports, pp. 3207--3242, DOI 10.4171/owr/2019/51. https://doi.org/10.4171/owr/2019/51. The registry separately records proposed year 2019; the relationship between this year and the citation's 2020 date is not explained.
• Supplied literature source: Colazzo--Jespers--Kubat--Van Antwerpen, "structure algebras." https://arxiv.org/abs/2305.06023. This abbreviated attribution is reproduced as supplied.
• Supplied literature source: "2026 structure-algebra results." https://doi.org/10.1016/j.jalgebra.2026.04.044. No authors or full article title are supplied for this reference.
• Supplied literature source: "original OWR report." https://doi.org/10.4171/owr/2019/51.
• Literature assessment date: 2026-08-22, reproduced exactly from literature_checked_at. This is supplied dated curation, not independently verified source checking. The supplied status is partially solved. The supplied assessment reports available prime-spectrum results and substantial domain, Noetherian, Gelfand--Kirillov dimension, and homological results for finite left-nondegenerate cases, while describing the full list of prime-image properties as unresolved. These claims and their applicability require independent verification; the restriction to finite left-nondegenerate cases in this assessment does not establish the intended scope of the original question.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review. These are supplied editorial assertions and do not establish independent mathematical verification of this draft or of the literature assessment.