Solomon-Terao algebra of hyperplane arrangements

We introduce a new algebra associated with a hyperplane arrangement $\mathcal{A}$, called the Solomon-Terao algebra $\mbox{ST}(\mathcal{A},\eta)$, where $\eta$ is a homogeneous polynomial. It is shown by Solomon and Terao that $\mbox{ST}(\mathcal{A},\eta)$ is Artinian when $\eta$ is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon-Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that $\mbox{ST}(\mathcal{A},\eta)$ is a complete intersection if and only if $\mathcal{A}$ is free. We also give a factorization formula of the Hilbert polynomials when $\mathcal{A}$ is free, and pose several related questions, problems and conjectures.