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
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