The Nodal Cubic is a Quantum Homogeneous Space

TL;DR

This paper constructs a specific Hopf algebra for the coordinate ring of the nodal cubic, demonstrating its structure as a quantum homogeneous space and exploring broader questions about affine varieties with this property.

Contribution

It provides the first explicit construction of a Hopf algebra making the coordinate ring of the nodal cubic a quantum homogeneous space.

Findings

The coordinate ring of the nodal cubic admits a Hopf algebra embedding as a right coideal subalgebra.

This construction supports the broader question of which affine varieties are quantum homogeneous spaces.

Abstract

The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring $B$ can be embedded as a right coideal subalgebra into a Hopf algebra $A$ such that $A$ is faithfully flat as a $B$-module. In the present article such a Hopf algebra $A$ is constructed for the coordinate ring $B$ of the nodal cubic, thus further motivating the question which affine varieties are quantum homogeneous spaces.