Let A and B be two connected graded commutative k-algebras of finite type, where k is a perfect field of positive characteristic p. We prove that the quasi--shuffle algebras generated by A and B are isomorphic as Hopf algebras if and only if A and B are isomorphic as graded k-vector spaces equipped with a Frobenius (pth-power) map.
For the hardest part of this analysis, we work with the dual construction, and are led to study connected graded cocommutative Hopf algebras H with two additional properties: H is free as an associative algebra, and the projection onto the indecomposables is split as a morphism of graded k-vector spaces equipped with a Verschiebung map.
Building on work on non-commutative Witt vectors by Goerss, Lannes, and Morel, we classify such free, `split' Hopf algebras.
A topological consequence is that, if X is a based path connected space, then the Hopf algebra H^*(Omega Sigma X;k) is determined by the stable homotopy type of X.
We also discuss the much easier analogous characteristic 0 results, and give a characterization of when our quasi--shuffle algebras are polynomial, generalizing the so-called Ditters conjecture.
Center problem, Abel equation and the Faa di Bruno Hopf algebra for output feedback
本文给出了非自治微分方程经典Poincare中心焦点问题的Devlin字问题的组合解释。从连通分次Hopf代数的观点出发，证明了Devlin的正则多项式与应用于Fferera的形式幂级数的霍普夫代数的对极的分次分量密切相关.这种联系是通过控制理论建立的，因为描述中心的阿贝尔方程等价于输出反馈方程，而输出反馈的霍普夫代数是由迭代积分的合成而不是迭代积分的乘积导出的，后者产生了洗牌代数。这意味着Devlin方法中起作用的主要代数结构实际上不是shuffle代数，而是Faa di Bruno类型的霍普夫代数，它是根据shuffle积定义的，但却是一个不同的代数结构。
