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

TL;DR

This paper generalizes known results about the antipode's square in connected graded Hopf algebras, extending to coalgebras over rings and reducing the exponent in key identities under certain conditions.

Contribution

It broadens the scope of antipode square identities to coalgebras over rings and introduces conditions for lowering the exponent in these identities.

Findings

Exponent n can be lowered to n-1 under specific conditions.

The condition (id - S^2)(H_2)=0 is crucial for exponent reduction.

Commutativity in H_1 elements suffices for the condition, as in FQSym.

Abstract

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 $\operatorname{FQSym}$.)