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

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