Quantum-symmetric equivalence for superpotential algebras

Hongdi Huang; Van C. Nguyen; Kent B. Vashaw; Padmini Veerapen; Xingting Wang

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arXiv · 2507.05612

Quantum-symmetric equivalence for superpotential algebras

Hongdi Huang·Van C. Nguyen·Kent B. Vashaw·Padmini Veerapen·Xingting Wang

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Abstract

We study superpotential algebras by introducing the notion of quantum-symmetric equivalence defined relatively to two fixed Hopf coactions. This concept relies on the non-vanishing of a bi-Galois object for the two coacting Hopf algebras, where the cotensor product with this object provides a Morita--Takeuchi equivalence between their comodule categories, mapping one superpotenial algebra to the other as comodule algebras. In particular, we investigate $\mathcal{GL}$-type and $\mathcal{SL}$-type quantum-symmetric equivalences using Bichon's reformation of bi-Galois objects in the language of cogroupoids constructed by nondegenerate twisted superpotentials. As applications, for the $\mathcal{GL}$-type, we characterize the Artin--Schelter regularity, or equivalently, twisted Calabi--Yau property, of a superpotential algebra as the non-vanishing of the bi-Galois object in the associated cogroupoid. For the $\mathcal{SL}$-type, we apply the pivotal structure of the comodule categories to study numerical invariants for $\mathcal{SL}$ quantum-symmetric equivalence, including the quantum Hilbert series of the superpotential algebras.

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BibTeX RIS

Hongdi Huang, Van C. Nguyen, Kent B. Vashaw, Padmini Veerapen, Xingting Wang. 2026-09-11. Quantum-symmetric equivalence for superpotential algebras. https://arxiv.org/abs/2507.05612

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