Quasi-homogeneity of potentials
type: Item
instance of: scholarly article
title: Quasi-homogeneity of potentials
zbMATH Open document ID: 1481.16032
DOI: 10.4171/JNCG/415
author: Zheng Hua
author: Guisong Zhou
published in: Journal of Noncommutative Geometry
publication date: 2022-02-08T00:00:00Z
review text: Summary: In noncommutative differential calculus, Jacobi algebra (or potential algebra) plays the role of Milnor algebra in the commutative case. The study of Jacobi algebras is of broad interest to researchers in cluster algebra, representation theory and singularity theory. In this article, we study the quasi-homogeneity of a potential in a complete free algebra over an algebraic closed field of characteristic zero. We prove that a potential with finite dimensional Jacobi algebra is right equivalent to a weighted homogeneous potential if and only if the corresponding class in the \(0\)th Hochschlid homology group of the Jacobi algebra is zero. This result can be viewed as a noncommutative version of the famous theorem of Kyoji Saito on isolated hypersurface singularities.
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 7471017
zbMATH Keywords: Jacobi algebra
zbMATH Keywords: quasi-homogeneous potential
zbMATH Keywords: Jordan-Chevalley decomposition
zbMATH Keywords: noncommutative differential calculus
MaRDI profile type: Publication
cites work: Cluster categories for algebras of global dimension 2 and quivers with potential.
cites work: The diamond lemma for ring theory
cites work: Gopakumar-Vafa invariants do not determine flops
cites work: Quivers with potentials and their representations. I: Mutations.
cites work: Noncommutative deformations and flops
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Classification of isolated hypersurface singularities by their moduli algebras
cites work: A cyclic derivative in noncommutative algebra
cites work: Quasihomogeneous isolated singularities of hyperplanes.
description: scientific article; zbMATH DE number 7471017