Hecke symmetries: an overview of Frobenius properties

TL;DR

This paper broadens the understanding of Frobenius properties in Hecke symmetries, removing previous restrictions on parameters and exploring their algebraic structures and automorphisms.

Contribution

It generalizes known results on Frobenius properties of R-symmetric and R-skewsymmetric algebras for Hecke symmetries without parameter restrictions.

Findings

R-skewsymmetric and R-symmetric algebras are Frobenius regardless of q.

An even Hecke symmetry yields two graded Frobenius algebras.

Artin-Schelter regular algebras of dimension 3 are not linked to quantum GL(3).

Abstract

This paper improves several previously known results. First, the results describing the R-skewsymmetric algebra and the quadratic dual of the R-symmetric algebra as Frobenius algebras are shown to be true with any restriction on the parameter q of the Hecke relation being removed. An even Hecke symmetry gives rise to a pair of graded Frobenius algebras. We describe interrelation between the Nakayama automorphisms of the two algebras. As an illustration of general technique we give full details of the verification that Artin-Schelter regular algebras of global dimension 3 and elliptic type A are not associated with any quantum GL(3).