Yang-Baxter deformations and rack cohomology

TL;DR

This paper classifies deformations of set-theoretic solutions to the Yang-Baxter equation derived from racks, especially in the modular case, showing they are gauge-equivalent to simpler quasi-diagonal forms and linking cohomology to rack cohomology.

Contribution

It provides a classification of Yang-Baxter deformations for racks in the modular case and connects Yang-Baxter cohomology to rack cohomology, revealing when deformations are trivial or non-trivial.

Findings

Deformations are gauge-equivalent to quasi-diagonal ones.

Yang-Baxter cohomology reduces to rack cohomology in the extreme case.

Non-trivial deformations exist in the modular case, linked to rack cohomology.

Abstract

Every rack $Q$ provides a set-theoretic solution $c_Q$ of the Yang-Baxter equation. This article examines the deformation theory of $c_Q$ within the space of Yang-Baxter operators over a ring $\A$, a problem initiated by Freyd and Yetter in 1989. As our main result we classify deformations in the modular case, which had previously been left in suspense, and establish that every deformation of $c_Q$ is gauge-equivalent to a quasi-diagonal one. Stated informally, in a quasi-diagonal deformation only behaviourally equivalent elements interact. In the extreme case, where all elements of $Q$ are behaviourally distinct, Yang-Baxter cohomology thus collapses to its diagonal part, which we identify with rack cohomology. The latter has been intensively studied in recent years and, in the modular case, is known to produce non-trivial and topologically interesting Yang-Baxter deformations.