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 S2 = id (where S2 = S ◦ 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≥0Hn, then each positive integer n satisfies(id −S2) n(Hn) = 0 and (even stronger) ((id +S) ◦(id −S2) n−1) (Hn) = 0. For 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 S2 by “coalgebra homomorphisms” (of sorts). Specializing back to connected graded Hopf algebras, we show that the exponent n in the identity( id −S2)n(Hn) = 0 can be lowered to n − 1 (for n > 1) if and only if(id −S2) (H2) = 0. (A sufficient condition for this is that every pair of elements of H1 commutes; this is satisfied, e.g., for FQSym.).