Orbifold Jacobian algebras for invertible polynomials

Abstract

An important invariant of a polynomial $f$ is its Jacobian algebra defined by its partial derivatives. Let $f$ be invariant with respect to the action of a finite group of diagonal symmetries $G$. We axiomatically define an orbifold Jacobian $\mathbb{Z}/2\mathbb{Z}$-graded algebra for the pair $(f,G)$ and show its existence and uniqueness in the case, when $f$ is an invertible polynomial. In case when $f$ defines an ADE singularity, we illustrate its geometric meaning.