Twisting Manin's universal quantum groups and comodule algebras

TL;DR

This paper explores quantum-symmetric equivalences of graded algebras via Morita-Takeuchi equivalences of their universal quantum groups, revealing invariance of key algebraic properties and classifying Koszul Artin-Schelter regular algebras.

Contribution

It introduces quantum-symmetric equivalence, studies its invariants, and characterizes 2-cocycle twists across various algebraic structures, unifying several concepts under this framework.

Findings

Invariance of homological and algebraic invariants under quantum-symmetric equivalence.

Koszul Artin-Schelter regular algebras of fixed global dimension form a single equivalence class.

Finite generation of Hochschild cohomology rings is preserved under certain 2-cocycle twists.

Abstract

We introduce the notion of quantum-symmetric equivalence of two connected graded algebras, based on Morita-Takeuchi equivalences of their universal quantum groups, in the sense of Manin. We study homological and algebraic invariants of quantum-symmetric equivalence classes, and prove that numerical $\mathrm{Tor}$-regularity, Castelnuovo-Mumford regularity, Artin-Schelter regularity, and the Frobenius property are invariant under any Morita-Takeuchi equivalence. In particular, by combining our results with the work of Raedschelders and Van den Bergh, we prove that Koszul Artin-Schelter regular algebras of a fixed global dimension form a single quantum-symmetric equivalence class. Moreover, we characterize 2-cocycle twists (which arise as a special case of quantum-symmetric equivalence) of Koszul duals, of superpotentials, of superpotential algebras, of Nakayama automorphisms of twisted Frobenius algebras, and of Artin-Schelter regular algebras. We also show that finite generation of Hochschild cohomology rings is preserved under certain 2-cocycle twists.