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

TL;DR

This paper introduces Yetter-Drinfeld post-Hopf algebras and explores their relationship with relative Rota-Baxter operators, extending the theory beyond cocommutative cases and connecting to existing algebraic structures.

Contribution

It defines Yetter-Drinfeld post-Hopf algebras, establishes their categorical equivalence with Yetter-Drinfeld braces and relative Rota-Baxter operators, and generalizes previous cocommutative results.

Findings

Yetter-Drinfeld post-Hopf algebras are isomorphic to Yetter-Drinfeld braces.

The category of Yetter-Drinfeld post-Hopf algebras is equivalent to a subcategory of relative Rota-Baxter operators.

The framework extends to non-cocommutative cases via an adjunction.

Abstract

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 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.