# On the graded algebras associated with Hecke symmetries

## TL;DR

This paper investigates the algebraic properties like Koszulness and Gorensteinness of quantum symmetric and intertwining algebras linked to Hecke symmetries, extending previous results to cases where q is a root of unity.

## Contribution

It extends the analysis of graded algebra properties associated with Hecke symmetries without restricting the parameter q, especially when q is a root of unity.

## Key findings

- Established conditions for Koszulness and Gorensteinness of these algebras.
- Identified restrictions on modules for Hecke algebras at roots of unity.
- Extended previous results to more general q values.

## Abstract

We consider quantum symmetric algebras, FRT bialgebras and, more generally, intertwining algebras for pairs of Hecke symmetries which represent quantum hom-spaces. The paper makes an attempt to investigate Koszulness and Gorensteinness of those graded algebras without a restriction on the parameter q of the Hecke relation used earlier. When q is a root of 1, positive results require a restriction on the indecomposable modules for the Hecke algebras of type A that can occur as direct summands of representations in the tensor powers of the base space.