On Separable and Frobenius Cowreaths of type $(A \otimes H^{\mathrm{op}}, H,\psi)$

Abstract

Cowreaths of type $(A \otimes H^{\mathrm{op}},H,\psi)$ have been investigated in [10,19-21,27] as examples of (h-)separable and Frobenius coalgebras in monoidal categories. They are entwining structures built from a Hopf algebra $H$ and an $H$-comodule algebra $(A,\rho_A)$. In this article we develop a general theory that allows us to recover results from the aforementioned papers in an easier way and also to extend them to cowreaths in higher dimension. Focusing on separability, the crucial observation is that, when $H$ has bijective antipode, one should work with integrals on the coalgebra $(H,\psi)$ in place of Casimir morphisms, for they are easier to classify. On the other hand, in dealing with Frobenius properties, we obtain that a cowreath $(A \otimes H^{\mathrm{op}},H,\psi)$ is Frobenius if and only if the morphism $(\mathrm{Id}_A \otimes \mu)\rho_A$ is inner ($\mu$ is the distinguished grouplike element in $H^*$). While Frobenius cowreaths are always (h-)separable, examples of (h-)separable cowreaths that are not Frobenius will be presented at the end of this article.