{"docId":10431,"paperId":10431,"url":"https:\/\/cm.episciences.org\/10431","doi":"10.46298\/cm.10431","journalName":"Communications in Mathematics","issn":"1804-1388","eissn":"2336-1298","volume":[{"vid":673,"name":"Volume 31 (2023), Issue 1"}],"section":[],"repositoryName":"arXiv","repositoryIdentifier":"2109.02101","repositoryVersion":3,"repositoryLink":"https:\/\/arxiv.org\/abs\/2109.02101v3","dateSubmitted":"2022-12-06 21:11:06","dateAccepted":"2022-12-06 21:22:53","datePublished":"2022-12-13 11:00:46","titles":["On the square of the antipode in a connected filtered Hopf algebra"],"authors":["Grinberg, Darij"],"abstracts":["It is well-known that the antipode $S$ of a commutative or cocommutative Hopf algebra satisfies $S^{2}=\\operatorname*{id}$ (where $S^{2}=S\\circ S$). Recently, similar results have been obtained by Aguiar, Lauve and Mahajan for connected graded Hopf algebras: Namely, if $H$ is a connected graded Hopf algebra with grading $H=\\bigoplus_{n\\geq0}H_n$, then each positive integer $n$ satisfies $\\left( \\operatorname*{id}-S^2\\right)^n \\left( H_n\\right) =0$ and (even stronger) \\[ \\left( \\left( \\operatorname{id}+S\\right) \\circ\\left( \\operatorname{id}-S^2\\right)^{n-1}\\right) \\left( H_n\\right) = 0. \\] For some specific $H$'s such as the Malvenuto--Reutenauer Hopf algebra $\\operatorname{FQSym}$, the exponents can be lowered. In this note, we generalize these results in several directions: We replace the base field by a commutative ring, replace the Hopf algebra by a coalgebra (actually, a slightly more general object, with no coassociativity required), and replace both $\\operatorname{id}$ and $S^2$ by \"coalgebra homomorphisms\" (of sorts). Specializing back to connected graded Hopf algebras, we show that the exponent $n$ in the identity $\\left( \\operatorname{id}-S^2\\right) ^n \\left( H_n\\right) =0$ can be lowered to $n-1$ (for $n>1$) if and only if $\\left( \\operatorname{id} - S^2\\right) \\left( H_2\\right) =0$. (A sufficient condition for this is that every pair of elements of $H_1$ commutes; this is satisfied, e.g., for $\\operatorname{FQSym}$.)","Comment: Published version. See v2 for a (slightly less terse) preprint"],"keywords":["Mathematics - Rings and Algebras","Mathematics - Combinatorics","Mathematics - Quantum Algebra","16T05, 16T30"]}