Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.05612 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916832013713408 |
|---|---|
| author | Huang, Hongdi Nguyen, Van C. Vashaw, Kent B. Veerapen, Padmini Wang, Xingting |
| author_facet | Huang, Hongdi Nguyen, Van C. Vashaw, Kent B. Veerapen, Padmini Wang, Xingting |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05612 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum-symmetric equivalence for superpotential algebras Huang, Hongdi Nguyen, Van C. Vashaw, Kent B. Veerapen, Padmini Wang, Xingting Quantum Algebra 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. |
| title | Quantum-symmetric equivalence for superpotential algebras |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2507.05612 |