On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra

Abstract

Recently the notion of post-Hopf algebra was introduced, with the universal enveloping algebra of a post-Lie algebra as the fundamental example. A novel property is that any cocommutative post-Hopf algebra gives rise to a sub-adjacent Hopf algebra with a generalized Grossman-Larson product. By twisting the post-Hopf product, we provide a combinatorial antipode formula for the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra. Relating to such a sub-adjacent Hopf algebra, we also obtain a closed inverse formula for the Guin-Oudom isomorphism in the context of post-Lie algebras. Especially as a byproduct, we derive a cancellation-free antipode formula for the Grossman-Larson Hopf algebra of ordered trees through a concrete tree-grafting expression.

21 pages

References in corpus (6)

• Integration and geometrization of Rota-Baxter Lie algebras
• Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation
• Rota-Baxter operators on cocommutative Hopf algebras
• Algebraic aspects of connections: from torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials
• A Survey on the Munthe-Kaas-Wright Hopf Algebra
• What is the Magnus Expansion?