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Involutive (simple) latin solutions of the Yang-Baxter equation and related (left) quasigroups

Marco Bonatto, Marco Castelli

TL;DR

This work advances the understanding of involutive, non-degenerate set-theoretic solutions to the Yang–Baxter equation by focusing on those with regular displacement groups and examining their block-imprimitivity and congruence structures. It develops a unified language via left quasigroups and cycle sets, connecting them to left braces and affine constructions, and provides concrete classifications for simple cycle sets, especially when the displacement group is nilpotent or regular. A key achievement is the complete description of simple cycle sets with prime-power size, the identification of the First Weyl Algebra as a natural ambient for affine simple cycle sets, and a detailed classification of simple affine cycle sets of minimal size $|X|=p^p$ in terms of irreducible representations of $A_1(\mathbb{Z}/p\mathbb{Z})$. The results yield explicit constructions, nonexistence results for certain cyclic-displacement cases, and general methods to derive complete block systems and congruences, enhancing both theoretical insight and practical generation of examples in the theory of set-theoretic Yang–Baxter solutions.

Abstract

In this paper, we study involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation with regular displacement group. In particular, we completely describe the blocks of imprimitivity and the congruences of the irretractable ones, that we show belonging to the class of the latin solutions. Among these solutions, we characterise the simple ones having nilpotent permutation group. A more precise description involving the First Weyl Algebra will be provided when the displacement group is abelian and normal in the total permutation group, and we enumerate and classify the simple ones having minimal size $p^p$, for an arbitrary prime number $p$. Finally, we illustrate our results by some examples.

Involutive (simple) latin solutions of the Yang-Baxter equation and related (left) quasigroups

TL;DR

This work advances the understanding of involutive, non-degenerate set-theoretic solutions to the Yang–Baxter equation by focusing on those with regular displacement groups and examining their block-imprimitivity and congruence structures. It develops a unified language via left quasigroups and cycle sets, connecting them to left braces and affine constructions, and provides concrete classifications for simple cycle sets, especially when the displacement group is nilpotent or regular. A key achievement is the complete description of simple cycle sets with prime-power size, the identification of the First Weyl Algebra as a natural ambient for affine simple cycle sets, and a detailed classification of simple affine cycle sets of minimal size in terms of irreducible representations of . The results yield explicit constructions, nonexistence results for certain cyclic-displacement cases, and general methods to derive complete block systems and congruences, enhancing both theoretical insight and practical generation of examples in the theory of set-theoretic Yang–Baxter solutions.

Abstract

In this paper, we study involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation with regular displacement group. In particular, we completely describe the blocks of imprimitivity and the congruences of the irretractable ones, that we show belonging to the class of the latin solutions. Among these solutions, we characterise the simple ones having nilpotent permutation group. A more precise description involving the First Weyl Algebra will be provided when the displacement group is abelian and normal in the total permutation group, and we enumerate and classify the simple ones having minimal size , for an arbitrary prime number . Finally, we illustrate our results by some examples.
Paper Structure (20 sections, 58 theorems, 41 equations)

This paper contains 20 sections, 58 theorems, 41 equations.

Table of Contents

  1. Left quasigroups
  2. Basics
  3. Coverings of indecomposable left quasigroups
  4. Blocks of imprimitivity
  5. Left braces
  6. Cycle sets
  7. Basics
  8. Indecomposable cycle sets of prime-power size
  9. Blocks of imprimitivity
  10. Cycle sets with regular displacement group
  11. Right linear cycle sets
  12. Right linear representation
  13. Simple cycle sets
  14. Simple left quasigroups
  15. Simple cycle sets with nilpotent displacement group
  16. ...and 5 more sections

Key Result

Lemma 1

AG If $X,Y$ are left quasigroups and $p:X\rightarrow Y$ is a left quasigroups epimorphism, the assignment $\sigma_x\mapsto \sigma_{p(x)}$ extends to a groups epimorphism $\bar{p}:\mathcal{G}(X)\rightarrow \mathcal{G}(Y)$.

Theorems & Definitions (118)

  • Lemma 1
  • Definition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 2
  • Definition 3
  • Proposition 4
  • proof
  • ...and 108 more