Quantum-symmetric equivalence is a graded Morita invariant

TL;DR

This paper establishes that graded Morita equivalence between m-homogeneous algebras implies quantum-symmetric equivalence, linking their universal quantum groups and showing that Zhang twists correspond to 2-cocycle twists within this framework.

Contribution

It proves that Morita equivalence of graded module categories induces quantum-symmetric equivalence, revealing a new invariant in the context of m-homogeneous algebras.

Findings

Morita equivalent m-homogeneous algebras are quantum-symmetrically equivalent.

Zhang twists of such algebras are 2-cocycle twists from their universal quantum groups.

Establishes a connection between graded Morita invariants and quantum group symmetries.

Abstract

We show that if two $m$-homogeneous algebras have Morita equivalent graded module categories, then they are quantum-symmetrically equivalent, that is, there is a monoidal equivalence between the categories of comodules for their associated universal quantum groups (in the sense of Manin) which sends one algebra to the other. As a consequence, any Zhang twist of an $m$-homogeneous algebra is a 2-cocycle twist by some 2-cocycle from its Manin's universal quantum group.