Yang–Baxter deformations and rack cohomology

Abstract:

In his study of quantum groups, Drinfeld suggested considering set-theoretic solutions of the Yang–Baxter equation as a discrete analogon. As a typical example, every conjugacy class in a group or, more generally, every rack $Q$ provides such a Yang–Baxter operator $c_Q \colon x \otimes y \mapsto y \otimes x^y$. In this article we study deformations of $c_Q$ within the space of Yang–Baxter operators over some complete ring. Infinitesimally these deformations are classified by Yang–Baxter cohomology. We show that the Yang–Baxter cochain complex of $c_Q$ homotopy-retracts to a much smaller subcomplex, called quasi-diagonal. This greatly simplifies the deformation theory of $c_Q$, including the modular case which had previously been left in suspense, by establishing that every deformation of $c_Q$ is gauge equivalent to a quasi-diagonal one. In a quasi-diagonal deformation only behaviourally equivalent elements of $Q$ interact; if all elements of $Q$ are behaviourally distinct, then the Yang–Baxter cohomology of $c_Q$ collapses to its diagonal part, which we identify with rack cohomology. This establishes a strong relationship between the classical deformation theory following Gerstenhaber and the more recent cohomology theory of racks, both of which have numerous applications in knot theory.