10.46298/cm.10431 https://cm.episciences.org/10431 Grinberg, Darij Darij Grinberg On the square of the antipode in a connected filtered Hopf algebra 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 episciences.org Mathematics - Rings and Algebras Mathematics - Combinatorics Mathematics - Quantum Algebra 16T05, 16T30 CC0 1.0 Universal (CC0 1.0) Public Domain Dedication 2022-12-06 2022-12-13 2022-12-13 eng journal article arXiv:2109.02101 10.48550/arXiv.2109.02101 2336-1298 https://cm.episciences.org/10431/pdf VoR application/pdf Communications in Mathematics Volume 31 (2023), Issue 1 Researchers Students