Yetter-Drinfeld post-Hopf algebras and Yetter-Drinfeld relative Rota-Baxter operators
Andrea Sciandra
TL;DR
The paper addresses extending post-Hopf algebras and relative Rota–Baxter operators beyond cocommutativity by introducing Yetter–Drinfeld post–Hopf algebras. It proves a sequence of deep correspondences: the category of Yetter–Drinfeld post–Hopf algebras is isomorphic to Yetter–Drinfeld braces (and thus to matched pairs of actions), and bijective Yetter–Drinfeld relative Rota–Baxter operators correspond to these structures with $R^{-1}$ forming a Yetter–Drinfeld 1–cocycle, thereby generalizing the cocommutative theories. The framework yields a subadjacent Hopf algebra $H_{\rightharpoonup}$ and shows that primitive elements carry a post–Lie structure, unifying post-Lie, brace, and RBO theories in the Yetter–Drinfeld setting. The adjunction persists without surjectivity, aligning with Li–Sheng–Tang in the cocommutative case. The work includes explicit non-cocommutative examples from coquasitriangular Hopf algebras (e.g., transmutations of $H_4$, $E(n)$, $SL_q(2)$, and $A_{1,2}^{\nu,\lambda}$), illustrating the breadth and utility of these generalized constructions.
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.