Hopf Algebra (Co)actions on Rational Functions

In the theory of quantum automorphism groups, one constructs Hopf algebras acting on an algebra K from certain algebra morphisms \( \sigma :K \rightarrow \textrm{M}_n(K)\) $\sigma :K\to {\text{M}}_n(K)$ . This approach is applied to the field \(K=k(t)\) $K=k(t)$ of rational functions, and it is investigated when these actions restrict to actions on the coordinate ring \(B=k[t^2,t^3]\) $B=k[t^2,t^3]$ of the cusp. An explicit example is described in detail and shown to define a new quantum homogeneous space structure on the cusp.