Classification of noncommutative central conics

TL;DR

This paper completes the classification of noncommutative central conics by establishing bijections among algebraic structures, advancing the understanding of noncommutative quadric hypersurfaces in algebraic geometry.

Contribution

It develops the general theory of homogenization and dehomogenization for noncommutative algebras to classify noncommutative central conics.

Findings

Bijections among isomorphism classes of 4-dimensional Frobenius algebras and noncommutative conics.

Complete classification of noncommutative central conics.

Framework connecting algebraic objects with geometric structures.

Abstract

Classification of noncommutative quadric hypersurfaces is one of the major projects in noncommutative algebraic geometry. In recent years, we are dedicated to complete the classification of noncommutative central conics. To achieve this goal, we and other authors develop some theories to study and classify some classes of noncommutative quadric hypersurfaces in a series of papers. Finally, in this paper, we completely classify noncommutative central conics by developing the general theory of homogenization and dehomogenization for noncommutative algebras and by previous results. As a main result, we show that there are bijections among the following sets of objects (i) the set of isomorphism classes of $4$-dimensional Frobenius algebras, (ii) the set of isomorphism classes of noncommutative affine pencils of conics, and (iii) the set of isomorphism classes of noncommutative central conics.