Quantum-symmetric equivalence for superpotential algebras

Abstract For two connected graded algebras, we introduce the notion of quantum-symmetric (QS) equivalence defined relative to two fixed Hopf coactions. In particular, we consider algebras associated to nondegenerate twisted superpotentials and investigate the $\mathcal{G}\mathcal{L}$-type and $\mathcal{S}\mathcal{L}$-type QS equivalences using Bichon’s reformulation of bi-Galois objects in the language of cogroupoids. For the $\mathcal{G}\mathcal{L}$-type, we characterize the Koszul Artin–Schelter (AS) regularity, or equivalently, the Koszul twisted Calabi–Yau (CY) property, of a superpotential algebra as the non-vanishing of the bi-Galois object in the associated cogroupoid. For the $\mathcal{S}\mathcal{L}$-type, we apply the pivotal structure of the comodule categories to study numerical invariants for $\mathcal{S}\mathcal{L}$ QS equivalence, including the quantum Hilbert series of the superpotential algebra.