Cofree compositions of coalgebras

Cofree compositions of coalgebras (extended abstract)

We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra with coalgebra map to D. We conclude by discussing these structures for compositions with bases the vertices of multiplihedra, composihedra, and hypercubes.

非对易变量的对称函数和拟对称函数的霍普夫代数是自由和余自由的

The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree

我们揭示了非交换变量的对称函数空间的结构，通过显示下划线的霍普夫代数是自由和co-free。引入了非交换变元拟对称函数的霍普夫代数，定义了该空间的单项式基上的积和余积，证明了该霍普夫代数是自由的和余自由的.在寻找生成空间的基的过程中，我们定义了集合划分和集合组合的顺序，这使我们能够定义具有简单而自然的规则的基元素的乘积。

置换与树的余交换霍普夫代数

Cocommutative Hopf algebras of permutations and trees

考虑Loday和Ronco的平面二叉树的霍普夫代数和Malvenuto和Alfretenauer的置换的余根滤子.我们给出了明确的同构，表明相关的分次霍普夫代数是对偶的上交换霍普夫代数在20世纪80年代末引进的Grossman和Larson。这些霍普夫代数分别由有序树和堆有序树构成。这些结果来自于这样一个事实，即只要从一个霍普夫代数出发，它是一个余自由分次余代数，则相应的分次霍普夫代数是一个shuffle霍普夫代数.

A propos de l'algèbre de Hopf des mots tassés WMat

In this article we study the packed words Hopf algebra WMat introduced by Duchamp, Hoang-Nghia et Tanasa. We start by explaining that WMat is not cofree, giving its antipode and describing its graded dual. We consider then a Hopf sub-algebra of permutations called $\mathfrak{S}\mathcal{H}$. Its graded dual $\mathfrak{S}\mathcal{H}^\circledast$ has a quadri-algebra structure, so it has a double dendriform algebra structure too. Thereafter, we introduce ISPW, a Hopf algebra of increasing strict packed words. It is graded, connected and cocommutative so is isomorphic to the enveloping algebra of …

Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here includ…

Hopf monoids, permutohedral cones, and generalized retarded functions

The commutative Hopf monoid of set compositions is a fundamental Hopf monoid internal to vector species, having undecorated bosonic Fock space the combinatorial Hopf algebra of quasisymmetric functions. We construct a geometric realization of this Hopf monoid over the adjoint of the (essentialized) braid hyperplane arrangement, which identifies the monomial basis with signed characteristic functions of the interiors of permutohedral tangent cones. We show that the indecomposable quotient Lie coalgebra is obtained by restricting functions to chambers of the adjoint arrangement, i.e. by quotient…