Plane curves which are quantum homogeneous spaces

TL;DR

This paper constructs specific quantum groups from plane curves defined by polynomial equations and explores their properties, conjecturing they contain the coordinate rings of these curves as quantum homogeneous spaces, with proofs in certain cases.

Contribution

It introduces new Hopf algebras associated with decomposable plane curves and proves they contain the curve's coordinate ring as a quantum homogeneous space in specific cases.

Findings

Constructed three new pointed Hopf algebras from plane curves.

Proved the conjecture for degrees up to 5 or when polynomials are powers of variables.

Connected the new algebras to known structures like downup algebras.

Abstract

Let $\mathcal{C}$ be a decomposable plane curve over an algebraically closed field $k$ of characteristic 0. That is, $\mathcal{C}$ is defined in $k^2$ by an equation of the form $g(x) = f(y)$, where $g$ and $f$ are polynomials of degree at least 2. We use this data to construct 3 pointed Hopf algebras, $A(x,a,g)$, $A(y,b,f)$ and $A(g,f)$, in the first two of which $g$ [resp. $f$] are skew primitive central elements, and the third being a factor of the tensor product of the first two. We conjecture that $A(g,f)$ contains the coordinate ring $\mathcal{O}(\mathcal{C})$ of $\mathcal{C}$ as a quantum homogeneous space, and prove this when each of $g$ and $f$ has degree at most 5 or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when $g$ has degree 3 $A(x,a,g)$ is a PBW deformation of the localisation at powers of a generator of the downup algebra $A(-1,-1,0)$.