Yang-Baxter deformations of quandles and racks

Given a rack Q and a ring A , one can construct a Yang-Baxter operator cQ : V ⊗ V → V ⊗ V on the free A-module V = AQ by setting cQ(x ⊗ y) = y ⊗ x y for all x, y ∈ Q . In answer to a question initiated by D.N.Yetter and P.J. Freyd, this article classifies formal deformations of cQ in the space of Yang-Baxter operators. For the trivial rack, where x = 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 cQ . In many cases this allows us to conclude that cQ 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. AMS Classification 17B37; 18D10,20F36,20G42,57M25