Symmetric algebras in categories of corepresentations and smash products (Q306524)

This paper studies Frobenius and symmetric algebras in the category \({\mathcal M}^H\) of right \(H\)-comodules over the finite dimensional Hopf algebra over the field \(k\). If \(A\) is a finite dimensional Frobenius algebra in this category, then the smash product \(A\# H^*\) is a Frobenius algebra in the category \({\mathcal M}^{H^*}\). An example is then given to show that the converse of this result is not true. Let now \(g\) and \(\alpha\) denote the distinguished grouplikes in \(H\) and \(H^*\). Assume that \(H\) is cosovereign by the character \(\alpha\), and also that \(H^*\) is cosovereign by \(g\). Then it is proved that \(A\) is a symmetric algebra in \({\mathcal M}^H\) with respect to \(\alpha\) if and only if \(A\# H^*\) is a symmetric algebra in \({\mathcal M}^{H^*}\) with respect to \(g\).