PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

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

Volume 255 / 2021

David Stanovský, Petr Vojtěchovský 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.

Authors

  • David StanovskýDepartment of Algebra
    Faculty of Mathematics and Physics
    Charles University
    Sokolovská 83
    18675 Praha, Czech Republic
    e-mail
  • Petr VojtěchovskýDepartment of Mathematics
    University of Denver
    2390 S. York St.
    Denver, CO 80208, U.S.A.
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image