报告题目：Post-Hopf algebras and post-Lie algebras: two adjunctions and two extensions

报告人：Andrea Sciandra

所在单位：Universit´e Libre de Bruxelles

报告时间：August 27, 2026, 21:00-23:00

报告地点：Zoom Id: 904 645 6677，Password: 2026

报告摘要:Post-Lie algebras were introduced by Vallette in an operadic context, and were later shown to arise from flat connections on smooth manifolds with covariantly constant torsion, from numerical integration on manifolds, and from relative Rota–Baxter operators of weight 1. Their Hopf-theoretic counterparts, postHopf algebras, were introduced by Li, Sheng and Tang, and in the cocommutative case they are equivalent to Hopf braces and to matched pairs of actions, hence they produce solutions of the quantum Yang–Baxter equation. Two adjunctions govern this picture: one between cocommutative post-Hopf algebras and relative Rota–Baxter operators on cocommutative Hopf algebras, and one given by the universal enveloping algebra and primitive elements functors between post-Lie algebras and post-Hopf algebras, the latter restricting to an equivalence on connected cocommutative objects (a post-analogue of the Cartier–Milnor–Moore theorem). In this talk, I will discuss two independent extensions of this picture. First, I will remove the cocommutativity assumption: replacing Hopf algebras by Hopf monoids in a category of Yetter–Drinfeld modules leads to Yetter–Drinfeld post-Hopf algebras and Yetter–Drinfeld relative Rota–Baxter operators, between which the first adjunction still holds; the corresponding structures are equivalent to matched pairs of actions on arbitrary Hopf algebras and produce solutions of the quantum Yang–Baxter equation in the noncocommutative setting, with interesting examples coming from coquasitriangular Hopf algebras via Majid’s transmutation. Second, I will discuss infinitesimal deformations of post structures, keeping the Lie bracket (respectively, the Hopf algebra structure) undeformed. This yields the notions of infinitesimal post-Lie algebra and infinitesimal post-Hopf algebra, for which the second adjunction, together with the Cartier–Milnor–Moore equivalence, is shown to lift. Along the way I will provide a classification of infinitesimal post-Lie structures onsl(2) and infinitesimal post-Hopf structures on Sweedler’s Hopf algebra, exhibit a Hochschild 2-cocycle on the subadjacent Hopf algebra in the cocommutative setting, and prove that the quadratic operad of infinitesimal post-Lie algebras is Koszul, by means of a filtered distributive law between the operads of Lie algebras and of bi-magmas. This second part is based on a recent joint work with A. Rivezzi and T. Weber

报告人简介：Andrea Sciandra is a postdoctoral researcher at the Universit´e Libre de Bruxelles. He obtained his PhD in December 2025 from the Universit`a di Torino and Politecnico di Torino, with a thesis on Hopf-algebraic structures related to the quantum Yang–Baxter equation. His research interests lie in abstract and categorical algebra, in particular Hopf algebras and their generalizations, skew braces, braided monoidal categories and semi-abelian categories