Idempotent solutions of the Yang-Baxter equation and twisted group division

Abstract

Idempotent left nondegenerate solutions of the Yang-Baxter equation are in one-to-one correspondence with twisted Ward left quasigroups, which are left quasigroups satisfying the identity $(x\ast y)\ast (x\ast z)=(y\ast y)\ast (y\ast z)$. Using combinatorial properties of the Cayley kernel and the squaring mapping, we prove that a twisted Ward left quasigroup of prime order is either permutational or a quasigroup. Up to isomorphism, all twisted Ward quasigroups $(X,\ast )$ are obtained by twisting the left division operation in groups (that is, they are of the form $x\ast y=\psi (x^{-1}y)$ for a group $(X,\cdot )$ and its automorphism $\psi$), and they correspond to idempotent latin solutions. We solve the isomorphism problem for idempotent latin solutions.