2025/07/08 by Hongdi Huang, Huang, Hongdi, Van C. Nguyen +7
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2507.05612
openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 GL-type and 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 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 SL-type, we apply the pivotal structure of the comodule categories to study numerical invariants for SL quantum-symmetric equivalence, including the quantum Hilbert series of the superpotential algebras.