Automorphisms of free braided associative algebras in two variables

TL;DR

This paper classifies the automorphism groups of two-variable free braided associative algebras with involutive diagonal braidings, revealing five distinct groups depending on the braiding form, and connects these to quantum plane automorphisms.

Contribution

It provides a complete description of automorphism groups for these algebras, extending understanding of their structure and relation to quantum planes.

Findings

Five different automorphism groups identified based on braiding form

Classification covers all automorphism groups of quantum planes

Results depend on the form of involutive diagonal braidings

Abstract

We describe the groups of automorphisms of two generated free braided associative algebras with involutive diagonal braidings over a field of characteristic $\neq 2$. Depending on the form of the diagonal involutive braiding, five different automorphism groups arise as automorphism groups of two generated free braided associative algebras. This list covers all three automorphism groups that arise in the case of quantum planes \cite{AC}.