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

TL;DR

This paper explores the structure of sub-adjacent Hopf algebras derived from post-Lie algebras, providing new formulas for antipodes and isomorphisms, with applications to combinatorial Hopf algebras of trees.

Contribution

It introduces a combinatorial antipode formula and inverse formula for sub-adjacent Hopf algebras of post-Lie algebras, advancing the understanding of their algebraic and combinatorial properties.

Findings

Derived a cancellation-free antipode formula for Grossman-Larson Hopf algebra.

Provided a closed inverse formula for Oudom-Guin isomorphism.

Established a generalized Grossman-Larson product in sub-adjacent Hopf algebras.

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 Oudom-Guin 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.