Classification of noncommutative conics associated to symmetric regular superpotentials

Abstract

Let [Formula: see text] be a [Formula: see text]-dimensional quantum polynomial algebra, and [Formula: see text] a central regular element. The quotient algebra [Formula: see text] is called a noncommutative conic. For a noncommutative conic [Formula: see text], there is a finite-dimensional algebra [Formula: see text] which determines the singularity of [Formula: see text]. In this paper, we mainly focus on a noncommutative conic such that its quadratic dual is commutative, which is equivalent to say, [Formula: see text] is determined by a symmetric regular superpotential. We classify these noncommutative conics up to isomorphism of the pairs [Formula: see text], and calculate the algebras [Formula: see text].