Defining relations of 3-dimensional cubic AS-regular algebras of Type P, S and T

2023/09/20 by Masaki Matsuno, Yu Saito, Matsuno, Masaki +1
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic structure #Algebraic structures and combinatorial models #Combinatorics #Homotopy and Cohomology in Algebraic Topology #Isomorphism (crystallography) #Mathematics #Morita equivalence #Noncommutative geometry #Pure mathematics #Superpotential

paper · pdf · doi:10.48550/arxiv.2309.11112

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2023/09/20 · openalex created_date 2023/09/22 · openalex updated_date 2026/07/28

Abstract

Classification of AS-regular algebras is one of the most important projects in noncommutative algebraic geometry. Recently, Itaba and the first author gave a complete list of defining relations of 3-dimensional quadratic AS-regular algebras by using the notion of geometric algebra and twisted superpotential. In this paper, we extend the notion of geometric algebra to cubic algebras, and give a geometric condition for isomorphism and graded Morita equivalence. One of the main results is a complete list of defining relations of 3-dimensional cubic AS-regular algebras corresponding to ℙ¹ × ℙ¹ or a union of irreducible divisors of bidegree (1,1) in ℙ¹ × ℙ¹. Moreover, we classify them up to isomorphism and up to graded Morita equivalence in terms of their defining relations.

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