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Deformations of Deligne's S_t

Primarily AI-generated textHuman understanding: some partsmath.QA — Quantum Algebra

Contributed by Noah Snyder ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

Snyder and Spencer showed that every *ribbon* infinitesimal deformation of the trivalent category St\mathcal{S}_t underlying Deligne's Rep(St)\mathrm{Rep}(S_t) comes from varying tt. Here we remove the braided hypothesis: every pivotal deformation of St\mathcal{S}_t, t≠0t\neq0, over a local Artinian C\mathbb{C}-algebra comes from varying tt. The pivotal hypothesis turns out to be automatic, so the same holds for monoidal deformations. Since St\mathcal{S}_t is not generated as a pivotal category by its trivalent vertex, the crossing must be added as a generator, and the Reidemeister moves only survive in a deformed form, with lower order terms in the 22- through 66-box spaces. An open condition on rotational eigenvalues lets us lift the crossing in a rotation invariant way. A handful of overlap computations, in the spirit of Liu's classification of Yang--Baxter relation planar algebras, then kill every lower order term up to a change of the chosen crossing.

Provenance statement

The first version of this note was written by Claude in response to the following prompt from Noah Snyder. Subsequent revisions are described in the paper. The prompt was accompanied by the arXiv versions of Snyder-Spencer arXiv:2609.19329 and Liu arXiv:1507.06030. "I recently wrote the attached paper which classifies *braided* deformations of Deligne's S_t. I think a similar idea should work without the braided assumption, but you'd need additional assumptions in the recognition theorem going up to the 6-box space in order to get deformed versions of the Reidemeister moves for trivalent graphs (the usual R1, R2, R3, but also the trivalent twist (R1.5) and moving the trivalent vertex through a crossing (R2.5)). These deformed relations will have some kind of lower order terms, but I think it should still be manageable along the lines of Liu's Yang-Baxter Relation classification (modified for the trivalent setting). Note that one of the key ideas of my paper is that adding rotational eigenvalue constraints (or similar ``open'' conditions) can let you get the deformation result without going through the harder combinatorics of the full classification. Can you prove that S\_t has no pivotal monoidal deformations following this outline?" I understand the outline and ideas of the approach well and I'm very confident that it will lead to a classification of pivotal deformations of S_t (though not that the answer would come out to be that there are no non-trivial ones!). I have not checked the calculation carefully. I have skimmed it and run it through a few tests.

Tools used

Anthropic
Claude OpusVersion 5.5

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Version history

  1. v1Submitted by Noah SnyderInitial depositCurrentOct 26, 2026