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

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. The latter structures are equivalent to matched pairs of actions on Hopf algebras and generalise both Hopf braces and Majid’s transmutation. We also prove that the category of Yetter—Drinfeld post-Hopf algebras is equivalent to a subcategory of Yetter—Drinfeld relative Rota—Baxter operators (that generalise bijective relative Rota—Baxter operators on cocommutative Hopf algebras). Once the surjectivity of the latter operators is removed, the equivalence is replaced by an adjunction and one recovers, in the cocommutative case, the result of Li, Sheng and Tang. The talk is partially based on a joint work with D. Ferri.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar