On the square of the antipode in a connected filtered Hopf algebra

Abstract

It is well-known that the antipode $S$ of a commutative or cocommutative Hopf algebra satisfies $S^2=\mathrm{id}$ (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 $H={⨁}_{n\ge 0}{H}_n$, then each positive integer $n$ satisfies ${\left(\mathrm{id}-S^2\right)}^n\left(H_n\right)=0$ 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 $\mathrm{FQSym}$, 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 $\mathrm{id}$ and $S^2$ by "coalgebra homomorphisms" (of sorts). Specializing back to connected graded Hopf algebras, we show that the exponent $n$ in the identity ${\left(\mathrm{id}-S^2\right)}^n\left(H_n\right)=0$ can be lowered to $n-1$ (for $n>1$) if and only if $\left(\mathrm{id}-S^2\right)\left(H_2\right)=0$. (A sufficient condition for this is that every pair of elements of $H_1$ commutes; this is satisfied, e.g., for $\mathrm{FQSym}$.)

Published version. See v2 for a (slightly less terse) preprint

References in corpus (4)

• Hopf Algebras in Combinatorics
• Hopf algebras, from basics to applications to renormalization
• The characteristic polynomial of the Adams operators on graded connected Hopf algebras
• From Hopf algebras to rough paths and regularity structures