Hecke symmetries associated with twisted polynomial algebras in 3 indeterminates

TL;DR

This paper studies Hecke symmetries on 3-dimensional vector spaces where the associated R-symmetric algebra is a twisted polynomial algebra, showing that all such symmetries are twists of a standard form, enabling classification.

Contribution

It proves that any Hecke symmetry with a twisted polynomial algebra structure is a twist of a canonical symmetry, facilitating the classification of these symmetries.

Findings

Any such Hecke symmetry is a twist of a standard symmetry.

The associated R-symmetric algebra is isomorphic to a twisted polynomial algebra.

The paper provides a classification framework for these symmetries.

Abstract

We consider Hecke symmetries on a 3-dimensional vector space with the associated R-symmetric algebra isomorphic to the polynomial algebra $k[x_1,x_2,x_3]$ twisted by an automorphism. The main result states that any such a Hecke symmetry is itself a twist of a Hecke symmetry with the associated R-symmetric algebra isomorphic to $k[x_1,x_2,x_3]$. This allows us to describe equivalence classes of such Hecke symmetries.