Given a rack and a ring , one can construct a Yang–Baxter operator on the free –module by setting for all . In answer to a question initiated by D N Yetter and P J Freyd, this article classifies formal deformations of in the space of Yang–Baxter operators. For the trivial rack, where for all , one has, of course, the classical setting of –matrices and quantum groups. In the general case we introduce and calculate the cohomology theory that classifies infinitesimal deformations of . In many cases this allows us to conclude that is rigid. In the remaining cases, where infinitesimal deformations are possible, we show that higher-order obstructions are the same as in the quantum case.
"Yang–Baxter deformations of quandles and racks." Algebr. Geom. Topol. 5 (2) 537 - 562, 2005. https://doi.org/10.2140/agt.2005.5.537