A Noncommutative Symmetric System over the Grossman-Larson Hopf Algebra of Labeled Rooted Trees

Grossman-Larson 特征的正性刚性及 Hoffman 根树图的 Kingman 面

Positivity Rigidity for Grossman-Larson Characters and the Kingman Face of the Hoffman Rooted-Tree Graph

本文分类了 Grossman-Larson Hopf 代数在根树基上的非负实特征，揭示其由 Kingman paintbox 参数化，并刻画了 Hoffman 根树图上的中心测度面及 Martin 边界结构。

NCS Systems over Differential Operator Algebras and the Grossman-Larson Hopf Algebras of Labeled Rooted Trees

Let $K$ be any unital commutative $\bQ$-algebra and $W$ any non-empty subset of $\bN^+$. Let $z=(z_1, ..., z_n)$ be commutative or noncommutative free variables and $t$ a formal central parameter. % Denote uniformly by $\kzz$ and $\kttzz$ the formal power series algebras % of $z$ over $K$ and $K[[t]]$, respectively. Let $\cDazz$ $(α\geq 1)$ be the unital algebra generated by the differential operators of $\kzz$ which increase the degree in $z$ by at least $α-1$ and $ \ataz $ the group of automorphisms $F_t(z)=z-H_t(z)$ of $\kttzz$ with $o(H_t(z))\geq α$ and $H_{t=0}(z)=0$. First, we study a co…

Combinatorics of Rooted Trees and Hopf Algebras

We begin by considering the graded vector space with a basis consisting of rooted trees, graded by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices. We define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the multiplicities associated with each operator. The symmetric algebra on the vector space of rooted trees (after a deg…

关于post-Lie代数的泛包络代数的次邻接Hopf代数

On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra

本文通过引入post-Hopf代数概念，研究post-Lie代数的泛包络代数的次邻接Hopf代数，利用扭结post-Hopf乘积给出组合反演公式，并得到Oudom-Guin同构的闭逆公式，进而导出有序树Grossman-Larson Hopf代数的无消去反演公式。

Rooted trees and symmetric functions: Zhao's homomorphism and the commutative hexagon

Recent work on perturbative quantum field theory has led to much study of the Connes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopf algebra of rooted trees, had already been studied by algebraists. L. Foissy introduced a noncommutative version of the Connes-Kreimer Hopf algebra, which turns out to be self-dual. Using some homomorphisms defined by the author and W. Zhao, we describe a commutative diagram that relates the aforementioned Hopf algebras to each other and to the Hopf algebras of symmetric functions, noncommutative symmetric functions, and quasi-symmetric functions…

Cocommutative Hopf algebras of permutations and trees

Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to the cocommutative Hopf algebras introduced in the late 1980's by Grossman and Larson. These Hopf algebras are constructed from ordered trees and heap-ordered trees, respectively. These results follow from the fact that whenever one starts from a Hopf algebra that is a cofree graded coalgebra, the associated graded Hopf algebra is a shuffle Hopf algebra.