A propos de l'alg\`ebre de Hopf des mots tass\'es WMat

TL;DR

This paper explores the structure of the Hopf algebra of packed words WMat, introduces sub-algebras and duals, and establishes isomorphisms with known algebraic structures like non-commutative symmetric functions.

Contribution

It provides a detailed analysis of WMat's properties, introduces the ISPW algebra, and links it to classical Hopf algebras such as QSym and NSym, revealing new structural insights.

Findings

WMat is not cofree, with its antipode and graded dual described.

The Hopf sub-algebra $rak{S}rak{H}$ has a quadri-algebra structure.

ISPW is isomorphic to non-commutative symmetric functions and related to extended compositions.

Abstract

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 its primitive elements. We describe some families of primitive elements. We prove that ISPW and non commutative symmetric functions are isomorphic. We define then an extended compositions Hopf algebra $\mathcal{C}_e$. It is not cocommutative but its primitive elements and those from ISPW are linked. We give an interpretation of $\mathcal{C}_e$ in terms of a semi-direct coproduct Hopf algebra. By using this, we can define two actions groups. We finish by giving an explicit isomorphism between ISPW$^\circledast$ and QSym and another one between ISPW and NSym.