Left-Garside categories, self-distributivity, and braids

2008/10/26 by Patrick Dehornoy, Dehornoy, Patrick
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR

paper · pdf · doi:10.48550/arxiv.0810.4698

arxiv created 2008/10/26 · openalex publication_date 2008/10/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In connection with the emerging theory of Garside categories, we develop the notions of a left-Garside category and of a locally left-Garside monoid. In this framework, the connection between the self-distributivity law LD and braids amounts to the result that a certain category associated with LD is a left-Garside category, which projects onto the standard Garside category of braids. This approach leads to a realistic program for establishing the Embedding Conjecture of [Dehornoy, Braids and Self-distributivity, Birkhauser (2000), Chap. IX].

Citations

• Garside structure for the braid group of G(e,e,r)
• Garside categories, periodic loops and cyclic sets
• Artin-Gruppen und Coxeter-Gruppen
• A classifying invariant of knots, the knot quandle
• Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups

Related

• Groupes de Garside
• Left regular representations of Garside categories I. C*-algebras and groupoids
• Shrinking braids and Left distributive monoid
• The 2-braid group and Garside normal form
• Garside-theoretic analysis of Burau representations