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

2024/08/02 by Li, Yunnan · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2408.01345

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.

Cited by

• An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence

Related

• The Cartier-Quillen-Milnor-Moore theorem in the Post-Hopf case
• On post-Lie structures for free Lie algebras
• An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence
• Pre- and Post-Lie Algebras: The Algebro-Geometric View
• Post-Lie algebras in Regularity Structures