Construction of a group of automorphisms for an infinite family of Garside groups

TL;DR

This paper constructs a specific group of automorphisms for an infinite family of Garside groups derived from solutions to the quantum Yang-Baxter equation, revealing new automorphism structures.

Contribution

It introduces a method to build automorphism groups for structure groups of symmetric solutions, preserving Garside properties and identifying outer automorphisms.

Findings

Automorphism groups are constructed for these Garside groups.

A subgroup preserving Garside properties is identified.

Outer automorphism groups are characterized in some cases.

Abstract

The structure groups of non-degenerate symmetric set-theoretical solutions of the quantum Yang-Baxter equation provide an infinite family of Garside groups with many interesting properties. Given a non-degenerate symmetric solution, we construct for its structure group a group of automorphisms. Moreover, we show this group of automorphisms admits a subgroup that preserves the Garside properties of the structure group. In some cases, we could also prove the group of automorphisms obtained is an outer automorphism group.