Garside structure on monoids with quadratic square-free relations

TL;DR

This paper explores the deep connections between Garside monoids, quadratic monoids with Frobenius Koszul duals, skew-polynomial monoids, and solutions to the Yang-Baxter equation, revealing underlying symmetries.

Contribution

It establishes new links between these algebraic structures, highlighting their combinatorial and algebraic symmetries and properties.

Findings

Identification of connections between Garside monoids and quadratic monoids

Demonstration of symmetry among various algebraic objects

Insights into the structure of solutions to the Yang-Baxter equation

Abstract

We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a "nice" Garside element, certain monoids $S$ with quadratic relations, whose monoidal algebra $A= k[S]$ has a Frobenius Koszul dual $A^{!}$ with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were introduced and studied by the author and in 1995 provided a new class of Noetherian Artin-Schelter regular domains, and the square-free set-theoretic solutions of the Yang-Baxter equation. There is a beautiful symmetry in these objects due to their nice combinatorial and algebraic properties.