Yetter–Drinfeld post-Hopf algebras and Yetter–Drinfeld relative Rota–Baxter operators

Recently, Li, Sheng, and Tang introduced post-Hopf algebras and relative Rota–Baxter operators (on cocommutative Hopf algebras), providing an adjunction between the respective categories under the assumption that the structures involved are cocommutative. We introduce Yetter–Drinfeld post-Hopf algebras, which become usual post-Hopf algebras in the cocommutative setting. In analogy with the correspondence between cocommutative post-Hopf algebras and cocommutative Hopf braces, the category of Yetter–Drinfeld post-Hopf algebras is isomorphic to the category of Yetter–Drinfeld braces introduced by the author in a joint work with D. Ferri. This allows to explore the connection with matched pairs of actions and provide examples of Yetter–Drinfeld post-Hopf algebras. Moreover, we prove that the category of Yetter–Drinfeld post-Hopf algebras is equivalent to a subcategory of bijective Yetter–Drinfeld relative Rota–Baxter operators. The latter structures coincide with the inverse maps of Yetter–Drinfeld 1-cocycles introduced by the author and D. Ferri and generalise bijective relative Rota–Baxter operators on cocommutative Hopf algebras. Hence, the previous equivalence passes to cocommutative post-Hopf algebras and bijective relative Rota–Baxter operators. Once the surjectivity of the Yetter–Drinfeld relative Rota–Baxter operators is removed, the equivalence is replaced by an adjunction and one can recover the result of Li, Sheng, and Tang in the cocommutative case.