Garside structure on monoids with quadratic square-free relations

2009/09/25 by Tatiana Gateva-Ivanova, Gateva-Ivanova, Tatiana
Computer Science · Mathematics · #16S36 #16S37 #16W35 #16W50 #81R50 #81R60 #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.0909.4709

openalex publication_date 2009/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Yang-Baxter maps: dynamical point of view
• "Set-theoretical" solutions of the quantum Yang-Baxter equation and a class of Garside groups

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• Quadratic algebras of skew type and the underlying semigroups
• The dual braid monoid
• Garside monoids vs divisibility monoids
• A dual braid monoid for the free group
• Groupes de Garside