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