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Structure groupoids of quiver-theoretic Yang-Baxter maps

Davide Ferri, Youichi Shibukawa

TL;DR

This work studies quiver-theoretic Yang–Baxter maps (YBMs) through the lens of structure categories and structure groupoids. It proves that the structure groupoid of an involutive non-degenerate YBM is a Garside groupoid, extending Chouraqui’s result from set-theoretic to quiver-theoretic YBMs by marrying weak RC-systems with Garside theory. The authors develop a framework linking YBMs to presented categories, establish a converse construction that yields YBMs from suitable presentations, and analyze principal homogeneous type solutions, including explicit PH-groupoid examples. The results provide new concrete Garside groupoids and deepen the interplay between YBMs, RC-systems, and enveloping groupoids, with implications for normal forms and the word problem in this rich algebraic setting.

Abstract

Solutions to the quiver-theoretic quantum Yang-Baxter equation are associated with structure categories and structure groupoids. We prove that the structure groupoids of involutive non-degenerate solutions are Garside. This generalises a well-known result about the structure groups of set-theoretic solutions, due to Chouraqui. We also construct involutive non-degenerate solutions from suitable presented categories. We then investigate the case of solutions of principal homogeneous type. Finally, we present some examples of this new class of Garside groupoids.

Structure groupoids of quiver-theoretic Yang-Baxter maps

TL;DR

This work studies quiver-theoretic Yang–Baxter maps (YBMs) through the lens of structure categories and structure groupoids. It proves that the structure groupoid of an involutive non-degenerate YBM is a Garside groupoid, extending Chouraqui’s result from set-theoretic to quiver-theoretic YBMs by marrying weak RC-systems with Garside theory. The authors develop a framework linking YBMs to presented categories, establish a converse construction that yields YBMs from suitable presentations, and analyze principal homogeneous type solutions, including explicit PH-groupoid examples. The results provide new concrete Garside groupoids and deepen the interplay between YBMs, RC-systems, and enveloping groupoids, with implications for normal forms and the word problem in this rich algebraic setting.

Abstract

Solutions to the quiver-theoretic quantum Yang-Baxter equation are associated with structure categories and structure groupoids. We prove that the structure groupoids of involutive non-degenerate solutions are Garside. This generalises a well-known result about the structure groups of set-theoretic solutions, due to Chouraqui. We also construct involutive non-degenerate solutions from suitable presented categories. We then investigate the case of solutions of principal homogeneous type. Finally, we present some examples of this new class of Garside groupoids.

Paper Structure

This paper contains 30 sections, 43 theorems, 125 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Scope and structure of the paper
  3. Preliminaries
  4. Quivers
  5. Generalities about categories
  6. Presented categories and presented groupoids
  7. Noetherianity
  8. Complemented presentations
  9. Enveloping groupoids and the Ore criterion
  10. Garside families
  11. Weak RC-systems and other cyclic systems
  12. Unit families and unital weak RC-systems
  13. Completions
  14. The structure category of a unital weak RC-system
  15. When the structure category is Garside
  16. ...and 15 more sections

Key Result

Lemma 2.16

Let ${\mathscr{C}}$ be a left-cancellative (resp. right-cancellative) category. If $x,y\in{\mathscr{C}}$ admit a right-lcm (resp. left-lcm) $z$, then every other right-lcm (resp. left-lcm) of $x$ and $y$ is $=^\times$-equivalent (resp. ${}^\times\!\!\!=$-equivalent) to $z$.

Figures (9)

  • Figure 1: A graphic interpretation of the relations ii and iii of Lemma \ref{['lem:extension-to-vartheta*']}, understood as consistency relations on a grid.
  • Figure 2: A graphic interpretation of the relations ii and iii of Lemma \ref{['lem:extension-to-vartheta*-dual']}, understood as consistency relations on a grid.
  • Figure 4: Cube relation for $\star$.
  • Figure 5: Cube relation for $\bullet$. The figure suggests that the argument for the proof of Proposition \ref{['prop:braided_quivers_are_w_co-RCs']}, is the proof of Proposition \ref{['prop:braided_quivers_are_wRCs']} with reversed arrows.
  • Figure 6: The definition of $\Delta_3(x_1, x_2, x_3)$$(\mathfrak{s}(x_1)=\mathfrak{s}(x_2)=\mathfrak{s}(x_3))$.
  • ...and 4 more figures

Theorems & Definitions (131)

  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.10
  • Definition 2.11
  • Definition 2.12
  • Definition 2.13
  • Definition 2.14
  • ...and 121 more