arXiv 2109.021012021-09math.RAmath.COmath.QA

关于连通滤余代数中反极元平方的注记

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

Darij Grinberg

备注：: Published version. See v2 for a (slightly less terse) preprint
发表信息：: Communications in Mathematics, Volume 31 (2023), Issue 1 (December 13, 2022) cm:10431
DOI：: 10.46298/cm.10431

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AI总结：

本文推广了连通分次 Hopf 代数中反极元平方的幂零性结果，在更一般的余代数框架下证明指数可降低的条件，并给出充分条件。

摘要：

众所周知，交换或余交换 Hopf 代数的反极元 $S$ 满足 $S^{2}=\operatorname*{id}$（其中 $S^{2}=S\circ S$）。最近，Aguiar、Lauve 和 Mahajan 对连通分次 Hopf 代数得到了类似结果：若 $H$ 是带有分次 $H=\bigoplus_{n\geq0}H_n$ 的连通分次 Hopf 代数，则每个正整数 $n$ 满足 $\left( \operatorname*{id}-S^2\right)^n \left( H_n\right) =0$，并且（更强地）\\[ \left( \left( \operatorname{id}+S\right) \circ\left( \operatorname{id}-S^2\right)^{n-1}\right) \left( H_n\right) = 0. \\] 对于某些特定的 $H$，例如 Malvenuto--Reutenauer Hopf 代数 $\operatorname{FQSym}$，指数可以降低。在本文中，我们在多个方向上推广这些结果：将基域替换为交换环，将 Hopf 代数替换为余代数（实际上是一个稍更一般的对象，不需要余结合性），并将 $\operatorname{id}$ 和 $S^2$ 都替换为（某种意义上的）“余代数同态”。特化回连通分次 Hopf 代数，我们证明在恒等式 $\left( \operatorname{id}-S^2\right) ^n \left( H_n\right) =0$ 中，指数 $n$ 可以降低到 $n-1$（当 $n>1$ 时）当且仅当 $\left( \operatorname{id} - S^2\right) \left( H_2\right) =0$。（一个充分条件是 $H_1$ 中任意两个元素交换；例如 $\operatorname{FQSym}$ 满足此条件。）

英文摘要：

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 $\operatorname{FQSym}$.)

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