On smash products of transitive module algebras. (Q606370)

Let \(k\) be a field of characteristic zero. Let \(H\) be a Hopf algebra over \(k\). An \(H\)-module algebra is called transitive if \(A^H=k1_A\) and \(A\) has no proper \(H\)-ideal. The main result of the paper says that if \(H\) is a finite-dimensional semisimple Hopf algebra over \(k\) and \(A\) is an \(s\)-dimensional transitive \(H\)-module algebra with a \(1\)-dimensional ideal \(k\lambda\), then \(A\#H\) is isomorphic to \(M_s(N)\), where \(N=\{h\in H\mid\sum h_{(1)}\cdot\lambda\otimes h_{(2)}=\lambda\otimes h\}\). This result is applied to the case of \(A=k(X)\), the function algebra on a finite set \(X\). The authors present counterexamples of finite-dimensional transitive \(H\)-module algebras over a non-semisimple Hopf algebra for which \(A\#H\) is non-isomorphic to \(M_{\dim A}(N)\).