Solomon-Terao algebra of hyperplane arrangements

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We introduce a new algebra associated with a hyperplane arrangement A, called the Solomon-Terao algebra ST(A,η), where η is a homogeneous polynomial. It is shown by Solomon and Terao that ST(A,η) is Artinian when η 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 ST(A,η) is a complete intersection if and only if A is free. We also give a factorization formula of the Hilbert polynomials when A is free, and pose several related questions, problems and conjectures.