Hecke symmetries associated with twisted polynomial algebras in 3 indeterminates

We consider Hecke symmetries on a 3-dimensional vector space with the associated R-symmetric algebra isomorphic to the polynomial algebra k[x₁,x₂,x₃] 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[x₁,x₂,x₃]. This allows us to describe equivalence classes of such Hecke symmetries.