Noncommutative Mather-Yau theorem and its applications to Calabi-Yau algebras

TL;DR

This paper extends the classical Mather-Yau theorem to noncommutative settings, showing that potentials are determined by their Jacobi algebras and Hochschild classes, with applications to Calabi-Yau algebras and cluster categories.

Contribution

It establishes a noncommutative analogue of the Mather-Yau theorem, linking potentials to their Jacobi algebras and Hochschild classes, advancing noncommutative singularity theory.

Findings

Potential equivalence is determined by Jacobi algebra and Hochschild class.

Right equivalence class determined by high jet of potential.

Applications to Ginzburg dg-algebras and cluster categories.

Abstract

In this article, we prove that for a finite quiver $Q$ the equivalence class of a potential up to formal change of variables of the complete path algebra $\widehat{\mathbb{C} Q}$, is determined by its Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential assuming the Jacobi algebra is finite dimensional. This is an noncommutative analogue of the famous theorem of Mather and Yau on isolated hypersurface singularities. We also prove that the right equivalence class of a potential is determined by its sufficiently high jet assuming the Jacobi algebra is finite dimensional. These two theorems can be viewed as a first step towards the singularity theory of noncommutative power series. As an application, we show that if the Jacobi algebra is finite dimensional then the corresponding complete Ginzburg dg-algebra, and the (topological) generalized cluster category thereof, are determined by the isomorphic type of the Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential.