We investigate the property of being Frobenius for some functors strictly related with Hopf modules over a bialgebra and how this property reflects on the latter. In particular, we characterize one-sided Hopf algebras with anti-(co)multiplicative one-sided antipode as those for which the free Hopf module functor is Frobenius. As a by-product, this leads us to relate the property of being an FH-algebra (in the sense of Pareigis) for a given bialgebra with the property of being Frobenius for certain naturally associated functors.

中文翻译：

---

Hopf 模块、Frobenius 函子和（单边）Hopf 代数

我们研究了一些与双代数上的 Hopf 模块严格相关的函子的 Frobenius 性质，以及该性质如何反映后者。特别地，我们将具有反（共）乘单边对映体的单边 Hopf 代数表征为自由 Hopf 模函子是 Frobenius 的那些。作为副产品，这导致我们将给定双代数的 FH 代数性质（在 Pareigis 的意义上）与某些自然关联函子的 Frobenius 性质联系起来。