Centrally Nilpotent Skew Braces of Maximal Class

Definitions and mathematical statement

A skew brace is a set $B$ with two group operations $+$ and $\circ$, with common identity $0$, satisfying \[ a\circ(b+c)=(a\circ b)-a+(a\circ c) \qquad(a,b,c\in B). \] The additive group need not be abelian. Write \[ \lambda_a(b)=-a+(a\circ b),\qquad a*b=\lambda_a(b)-b,\qquad [a,b]_+=a+b-a-b. \] For subsets $X,Y\subseteq B$, interpret $X*Y$ and $[X,Y]_+$ as the additive subgroups generated by the corresponding elementwise expressions. Interpret a sum of such terms as the additive subgroup generated by their union. These conventions make the supplied recurrence explicit; the extracted source does not specify its subgroup-generation or ideal-closure conventions, which require confirmation.

Set $\Gamma_1(B)=B$ and, for every integer $n\geq1$, use the recurrence supplied in the clean statement: \[ \Gamma_{n+1}(B)= \sum_{i=1}^{n}\left( \Gamma_i(B)*\Gamma_{n+1-i}(B) +[\Gamma_i(B),\Gamma_{n+1-i}(B)]_+ \right). \] For this specified series, a nonzero brace has class $c\geq1$ if \[ \Gamma_{c+1}(B)=\{0\},\qquad \Gamma_c(B)\neq\{0\}. \] Assign class $0$ to the one-element brace. The intended objects are centrally nilpotent skew braces, with the source proposing the class measured by this series. The source's assertion that this series is equivalent to an earlier central series requires verification because that earlier series is corrupted in the supplied extraction.

Requested classification, with an unresolved parameter. Study and classify centrally nilpotent skew braces having maximal class measured by the displayed series. To make "maximal" precise, a comparison family $\mathcal C$ must first be specified. Within such a family, a centrally nilpotent brace $B\in\mathcal C$ has maximal class when \[ c(B)\geq c(A) \quad\text{for every centrally nilpotent }A\in\mathcal C, \] provided a greatest class exists. Classification up to skew-brace isomorphism means classification under bijections preserving both group operations; this is a conventional interpretation of the requested classification, not an explicitly supplied equivalence relation. The record does not specify $\mathcal C$, a fixed order, a prime, or an extremal class bound. Consequently, it does not determine a unique precise maximal-class classification problem.

The extracted original also gives the purported terminating recurrence \[ \Gamma_{n+1}(B)=B*\Gamma_n(B)+\Gamma_n(B)+[B,\Gamma_n(B)]_+. \] As written, its right-hand side contains $\Gamma_n(B)$; with $\Gamma_1(B)=B$, it cannot terminate at $\{0\}$ for a nonzero brace. No missing term is reconstructed here. The source further reports, for finite skew braces, equivalence of central nilpotency with simultaneous left and right nilpotency, and notes the existence of non-trivial braces of order $p^2$. These are supplied contextual claims requiring source review; neither a finite-order restriction on the classification nor a meaning of "non-trivial" is fixed by them. Factorizations of skew braces and decompositions of set-theoretic Yang--Baxter solutions are supplied motivation, without a separate precise classification target.

Short English statement

Classify centrally nilpotent skew braces whose class is maximal for the displayed central series. The intended comparison family and maximal-class bound are missing, and an earlier recurrence is corrupted. This is therefore a statement-only draft with unresolved mathematical qualifications requiring human review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005441$, source identifier catalogue entry, title Centrally Nilpotent Skew Braces of Maximal Class. Repository registry references: data/unsolvedmath/problems.json and catalog/problems.json. No independent inspection of those files is asserted.
• Mini-Workshop: Skew Braces and the Yang--Baxter Equation (2023), Oberwolfach Reports, pp. 537--563, DOI 10.4171/owr/2023/9. Supplied original source and literature-source link: https://doi.org/10.4171/owr/2023/9. The extracted original labels this Problem 4. No individual proposer is supplied.
• Bonatto--Jedlička, Central nilpotency of skew braces. Supplied literature-source link: https://arxiv.org/abs/2109.04389.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem partially_solved, reports development of central nilpotency and its lower and upper series and structural results for several low classes, and reports that a classification at maximal class remains absent. This is supplied dated curation, not independently verified here.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked directly. This is a supplied source-verification claim, not a claim of verification in this draft. The original extraction contains a reference labeled [2], whose bibliographic identity is not supplied explicitly.