Yang–Baxter deformations of quandles and racks

Abstract

Given a rack $Q$ and a ring $\mathbb{A}$, one can construct a Yang–Baxter operator $c_Q:V\otimes V\to V\otimes V$ on the free $\mathbb{A}$–module $V=\mathbb{A}Q$ by setting $c_Q\left(x\otimes y\right)=y\otimes x^y$ for all $x,y\in Q$. In answer to a question initiated by D N Yetter and P J Freyd, this article classifies formal deformations of $c_Q$ in the space of Yang–Baxter operators. For the trivial rack, where $x^y=x$ for all $x,y$, one has, of course, the classical setting of $r$–matrices and quantum groups. In the general case we introduce and calculate the cohomology theory that classifies infinitesimal deformations of $c_Q$. In many cases this allows us to conclude that $c_Q$ 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.