Source and significance assessment
Checked September 9, 2026. One distinct proposed research contribution.
Published problem
Hani Abdelwahab, Ivan Kaygorodov, and Bauyrzhan Sartayev, Shift associative algebras, Rendiconti del Circolo Matematico di Palermo Series 2 74, article 145 (2025), published June 30, 2025, DOI and publisher text. The final institutional PDF was also inspected. Immediately after Proposition 64, on PDF page 25, the paper asks:
Is there a rigid non-associative shift associative algebra?
The definition is (ab)c=b(ca), and geometric rigidity means an open orbit in the entire variety of products satisfying the identity. Corollary 58 and Proposition 59 establish that nonassociativity first occurs in dimension five. Proposition 59 already gives a five-dimensional nonassociative multiplication table. Existence in dimension five by itself is therefore not a new contribution here.
Contribution and scope
P17 supplies a nilpotent example and proves its rigidity by equality of the full infinitesimal-deformation and orbit tangent spaces. An integer minor fixes the computational proof obligation. The orbit closure is a component consisting entirely of algebras with all fourfold products zero. The source's Vergne, Grunewald–O'Halloran, and combined properties consequently fail in dimension five. A proof over the dual numbers shows that adjoining scalar factors preserves infinitesimal rigidity, producing examples in all dimensions at least five.
These results form one paper. Neither the extension nor the degeneration corollaries are counted separately. The result is in finite-dimensional nonassociative algebra, with concrete uses for deformation calculations and subsequent geometric classification. It does not settle a Lie-algebra rigidity conjecture, classify all five-dimensional shift-associative algebras, or produce nilpotent rigid examples in every higher dimension.
Follow-up search and priority limits
Queries included combinations of “shift associative”, “nearly associative”, “rigid”, “rigidity”, “five”, “5-dimensional”, “2025”, and “2026”, including the exact published open-question wording. The inspected results did not identify an earlier resolution. This is a bounded search, not proof of exhaustive historical priority.
Relevant records inspected:
- Barreiro, Benayadi, Rizzo, Nearly associative algebras, Journal of Algebra 658 (2024), 821–868. Bibliography and abstract checked; the structural and quadratic-algebra results precede the explicit 2025 question.
- Hassan Alhussein, On the Cohomology of Cyclic Associative Algebras, June 29, 2026. Abstract inspected. This addresses the associative cyclic subclass; no resolution of the nonassociative rigidity question is asserted in the abstract.
- Barreiro and coauthors, On anti-left-invariant and left-invariant pseudo-Euclidean nearly associative algebras, August 2, 2026. Abstract inspected. It describes double-extension and metric-structure results; no rigidity resolution is asserted there. The full paper was not audited.
As an additional bounded check, both 2026 full HTML texts were opened and searched for the stem “rigid”; neither returned a match. This is a targeted text check, not full proof review.
The two 2026 papers are not mathematical dependencies of P17. The search did not establish that no related result exists elsewhere or inside an uninspected full text. A referee or specialist may identify prior art. No citation-count prediction is warranted.
Significance assessment
An explicit question in a recent peer-reviewed structural/classification paper is stronger evidence of research relevance than a self-generated conjecture. Minimal dimension, rigidity in the unrestricted variety, a nilpotent component, and the dimension-independent extension provide concrete later uses. The niche is specialized, however, and its eventual citation impact is uncertain. This is a fully written and internally checked proposed solution, not an externally certified publication-ready theorem.