  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.
