Yang–baxter Deformations and Rack Cohomology

Abstract. In his study of quantum groups, Drinfeld suggested to consider 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 cQ : x ⊗ y 7→ y ⊗ x . In this article we study deformations of cQ within the space of Yang–Baxter operators. Over a complete ring these are classified by Yang–Baxter cohomology. We show that the general Yang–Baxter cohomology complex of cQ homotopyretracts to a much smaller subcomplex, called quasi-diagonal. This greatly simplifies the deformation theory of cQ, including the modular case which had previously been left in suspense, by establishing that every deformation of cQ 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 cQ 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.