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arXiv · 2408.01345

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

Yunnan Li
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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.

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BibTeXRIS

Yunnan Li. 2024-08-02. On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra. https://doi.org/10.1016/j.jalgebra.2026.05.011

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