Pointed Hopf algebra (co)actions on rational functions

TL;DR

This paper develops a framework for constructing Hopf algebra actions on rational function fields, enabling the analysis of their properties and applications to quantum homogeneous spaces on algebraic curves.

Contribution

It introduces a method to build Hopf algebra actions on rational functions using algebra morphisms, and applies this to extend quantum homogeneous space structures to rational functions.

Findings

Constructed Hopf algebra actions on rational function fields.

Extended quantum homogeneous space structure from regular to rational functions.

Provided explicit example on the cusp curve.

Abstract

This article studies the construction of Hopf algebras $H$ acting on a given algebra $K$ in terms of algebra morphisms $ \sigma \colon K \rightarrow \mathrm{M}_n(K)$. The approach is particularly suited for controlling whether these actions restrict to a given subalgebra $B$ of $K$, whether $H$ is pointed, and whether these actions are compatible with a given $*$-structure on $K$. The theory is applied to the field $K=k(t)$ of rational functions containing the coordinate ring $B=k[t^2,t^3]$ of the cusp. An explicit example is described in detail and shown to define a quantum homogeneous space structure on the cusp, which, unlike the previously known one, extends from regular to rational functions.