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 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.
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论后李代数的普遍包络代数的子邻接霍普夫代数
最近,人们引入了后霍普夫代数的概念,并以后李代数的普遍展开代数作为基本例子。一个新颖的性质是,任何可交换后霍普夫代数都会产生具有广义格罗斯曼-拉森积的下相邻霍普夫代数。通过扭曲后霍普夫乘积,我们为后列代数的普遍包络代数的次相邻霍普夫代数提供了一个组合反求公式。关于这样的子邻接霍普夫代数,我们还得到了后列代数背景下的奥多姆-古因同构的封闭逆公式。特别是作为副产品,我们通过一个离散的树嫁接表达式,为有序树的格罗斯曼-拉森霍普夫代数推导出了一个无取消的反节点公式。
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