Idempotent solutions of the Yang–Baxter equation and twisted group division
Volume 255 / 2021
Fundamenta Mathematicae 255 (2021), 51-68
MSC: 16T25, 20N05, 20N02.
DOI: 10.4064/fm872-2-2021
Published online: 12 April 2021
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 $(x*y)*(x*z)=(y*y)*(y*z)$. 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 $(X,*)$ are obtained by twisting the left division operation in groups (that is, they are of the form $x*y=\psi (x^{-1}y)$ 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.
