Twisting Manin's universal quantum groups and comodule algebras

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 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 quantumsymmetric equivalence class. Moreover, we characterize 2co cycle twists (which arise as a special case of quantumsymmetric 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 -co cycle twists. (c) 2024 Elsevier Inc. All rights reserved.