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Primitive elements in infinitesimal bialgebras

Dalia Artenstein, Ana González, María Ronco

TL;DR

The paper studies unital $S$-magmatic infinitesimal bialgebras, focusing on free $S$-magmatic algebras built from colored planar rooted binary trees and endowed with a unique coassociative coproduct that satisfies the unital infinitesimal relation. It generalizes Aguiar–Sottile’s Tamari-order description to colored trees, introducing a Moebius-basis construction $M_t$ and a graded product $\rightthreetimes$ to describe primitive elements: the primitive space is generated by $M_t$ for $t$ $\rightthreetimes$-irreducible, and the entire algebra is cotensor over this primitive space. The framework is instantiated with the integer-relations Hopf algebra of Pilaud–Pons, where the dual coalgebra ${\mathbb H}_{{\mathcal{R}}_{PP}}^{*}$ is generated by a family of magmatic products $*_{\alpha}$ tied to reflexive relations, and the primitive subspace is explicitly described by $\sqcup$-irreducible relations via the map $\Xi$. Overall, the work provides a structural description of primitive elements in a broad class of magmatic infinitesimal bialgebras and demonstrates concrete realizations in a combinatorial Hopf-algebra setting.

Abstract

For any set S, the free magmatic algebra spanned by card(S) binary products is the vector space spanned by the set of all planar rooted binary trees with the internal nodes colored by the elements of S, graded by the number of leaves of a tree. We show that it has a unique structure of coassociative coalgebra such that the coproduct satisfies the unital infinitesimal condition with each magmatic product, and prove an analog of Aguiar-Sottile formula in this context, describing the coproduct in terms of the Moebius basis for the Tamari order. The last result allows us to compute the subspace of primitive elements of any unital infinitesimal S-magmatic bialgebra. As an example, we construct a set of generators of the dual of Pilaud and Pons bialgebra of integer relations and compute an explicit basis of its subspace of primitive elements.

Primitive elements in infinitesimal bialgebras

TL;DR

The paper studies unital -magmatic infinitesimal bialgebras, focusing on free -magmatic algebras built from colored planar rooted binary trees and endowed with a unique coassociative coproduct that satisfies the unital infinitesimal relation. It generalizes Aguiar–Sottile’s Tamari-order description to colored trees, introducing a Moebius-basis construction and a graded product to describe primitive elements: the primitive space is generated by for -irreducible, and the entire algebra is cotensor over this primitive space. The framework is instantiated with the integer-relations Hopf algebra of Pilaud–Pons, where the dual coalgebra is generated by a family of magmatic products tied to reflexive relations, and the primitive subspace is explicitly described by -irreducible relations via the map . Overall, the work provides a structural description of primitive elements in a broad class of magmatic infinitesimal bialgebras and demonstrates concrete realizations in a combinatorial Hopf-algebra setting.

Abstract

For any set S, the free magmatic algebra spanned by card(S) binary products is the vector space spanned by the set of all planar rooted binary trees with the internal nodes colored by the elements of S, graded by the number of leaves of a tree. We show that it has a unique structure of coassociative coalgebra such that the coproduct satisfies the unital infinitesimal condition with each magmatic product, and prove an analog of Aguiar-Sottile formula in this context, describing the coproduct in terms of the Moebius basis for the Tamari order. The last result allows us to compute the subspace of primitive elements of any unital infinitesimal S-magmatic bialgebra. As an example, we construct a set of generators of the dual of Pilaud and Pons bialgebra of integer relations and compute an explicit basis of its subspace of primitive elements.
Paper Structure (12 sections, 26 theorems, 82 equations)

This paper contains 12 sections, 26 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Aguiar-Sottile formula
  3. Unital augmented coalgebras
  4. Planar rooted trees
  5. Aguiar-Sottile' s formula in ${\mathbb K}[{\hbox{\bf PBT}}]$
  6. Magmatic Infinitesimal bialgebras
  7. Free $S$-magmatic infinitesimal algebras
  8. Primitive elements and Aguiar-Sottile' s formula
  9. An example: the coalgebra of integer relations
  10. The Hopf algebra ${\mathbb H}_{{\mathcal{R}}_{PP}^*}$
  11. Magmatic binary products generating the vector space ${\mathbb K}[{\mathcal{R}}]$
  12. Primitive elements in $({\mathbb H}_{{\mathcal{R}}_{PP}}^*, \Delta_{\mathcal{R}})$

Key Result

Lemma 2.2.10

Let $t$ and $w$ be a pair of trees in ${\hbox{\bf PBT}_n}$ such that $t = t_0{\underline{\circ}} (t_1,\dots ,t_r)$ and $w = w_0{\underline{\circ}} (w_1,\dots, w_r)$ for trees $t_0, w_0 \in {\hbox{\bf PBT}_r}$ and $t_i, w_i\in {\hbox{\bf PBT}_{n_i}}$, for $1\leq i\leq r$ and $\sum_{i=1}^r n_i = n$.

Theorems & Definitions (86)

  • Definition 2.1.1
  • Definition 2.1.3
  • Definition 2.1.4
  • Definition 2.2.1
  • Example 2.2.3
  • Definition 2.2.4
  • Remark 2.2.5
  • Definition 2.2.6
  • Definition 2.2.7
  • Definition 2.2.8
  • ...and 76 more