Involutive Yang–Baxter: cabling, decomposability, and Dehornoy class

  • Victoria Lebed

    Normandie Univ, UNICAEN, CNRS, LMNO, Caen, France
  • Santiago Ramírez

    IMAS–CONICET, Universidad de Buenos Aires, Argentina
  • Leandro Vendramin

    Vrije Universiteit Brussel, Belgium
Involutive Yang–Baxter: cabling, decomposability, and Dehornoy class cover
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Abstract

We develop new machinery for producing decomposability tests for involutive solutions to the Yang–Baxter equation. It is based on the seminal decomposability theorem of Rump and on “cabling” operations on solutions and their effect on the diagonal map . Our machinery yields an elementary proof of a recent decomposability theorem of Camp-Mora and Sastriques, as well as original decomposability results. It also provides a conceptual interpretation (using the language of braces) of the Dehornoy class, a combinatorial invariant naturally appearing in the Garsidetheoretic approach to involutive solutions.

Cite this article

Victoria Lebed, Santiago Ramírez, Leandro Vendramin, Involutive Yang–Baxter: cabling, decomposability, and Dehornoy class. Rev. Mat. Iberoam. (2023), published online first

DOI 10.4171/RMI/1438