In this paper we describe Hopf algebras which are associated with certain families of trees. These Hopf algebras originally arose in a natural fashion: one of the authors [5] was investigating data structures based on trees, which could be used to efficiently compute certain differential operators. Given data structures such as trees which can be multiplied, and which act as higherorder derivat...

Let CRF S denote the category of S-colored rooted forests, and HCRFS denote its Ringel-Hall algebra as introduced in [6]. We construct a homomorphism from a K+ 0 (CRF S)–graded version of the Hopf algebra of noncommutative symmetric functions to HCRFS . Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a K+ 0 (CRF S)–graded version of the algebra of quasisymmetric func...

We show that the differential complex Ω B over the braided matrix algebra BM q (N) represents a covariant comodule with respect to the coaction of the Hopf algebra Ω A which is a differential extension of GL q (N). On the other hand, the algebra Ω A is a covariant braided comodule with respect to the coaction of the braided Hopf algebra Ω B. Geometrical aspects of these results are discussed.

in this paper we initiate the study of real group algebras and investigate some of its aspects.let l1 (g) be a group algebra of a locally compact group g,τ :g →g be a group homeomorphismsuch that τ 2 =τοτ = 1, the identity map, and lp (g,τ ) = { f ∈ lp (g) : fοτ = f } ( p ≥ 1) . in thispaper, among other results, we clarify the structure of lp (g,τ ) and characterize amenability ofl1 (g,τ ) and...

We analyze the structure of the Malvenuto-Reutenauer Hopf algebraSSym of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. In addition, we describe the structure constants of the...

We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e. Baxter permutations, pairs of twin binary trees, etc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like ...

We introduce the Hopf algebra of uniform block permutations and show that it is self-dual, free, and cofree. These results are closely related to the fact that uniform block permutations form a factorizable inverse monoid. This Hopf algebra contains the Hopf algebra of permutations of Malvenuto and Reutenauer and the Hopf algebra of symmetric functions in non-commuting variables of Gebhard, Ros...

These notes are an extended version of a series of lectures given at Bogota from 2nd to 6th december 2002. They aim to present a self-contained introduction to the Hopf-algebraic techniques which appear in the work of A. Connes and D. Kreimer on renormalization in Quantum Field Theory [CK1], [CK2], [BF]... Our point of view consists in revisiting a substantial part of their work in the abstract...