Pointed Hopf algebra (co)actions on rational functions

Abstract

This article studies the construction of Hopf algebras $H$ acting on a given algebra $K$ in terms of algebra morphisms $\sigma :K\to {\mathrm{M}}_n(K)$. 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 $\ast$-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.

35 pages