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 = id (where S 2 = S • S). Recently, similar results have been obtained by Aguiar and Lauve for connected graded Hopf algebras: Namely, if H is a connected graded Hopf algebra with grading HFor some specific H's such as the Malvenuto-Reutenauer Hopf algebra 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 id and S 2 by "coalgebra homomorphisms" (of sorts). Specializing back to connected graded Hopf algebras, we show that the exponent n in Aguiar's identity id −S 2 n (H n ) = 0 can be lowered to n − 1 (for n > 1) if and only if id −S 2 (H 1 ) = 0.