Plane curves which are quantum homogeneous spaces

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 $A(-1,-1,0)$.