Quasi-homogeneity of potentials

Abstract

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 0th 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.