Duality between Quasi-Symmetrical Functions and the Solomon Descent Algebra

Malvenuto, Claudia; Reutenauer, Christophe

doi:10.1006/jabr.1995.1336

Cited by 371 publications

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“…The descent algebra carries naturally a Hopf algebra structure [25], [33]. Since D is freely generated by the p n , the coproduct is entirely defined by the requirement that the p n form a sequence of divided powers (that is, .p n / D i Cj Dn p i˝pj ) or, equivalently, that any Lie idempotent is a primitive element, see e.g.…”

Section: Lie Idempotents Actions On Pro-unipotent Groupsmentioning

confidence: 99%

A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion

Ebrahimi‐Fard¹,

Manchon²,

Patras³

2009

J. Noncommut. Geom.

103

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Abstract. The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here a theory of functional identities for noncommutative Rota-Baxter algebras which is shown to encode, among others, this process in the context of Connes-Kreimer's Hopf algebra of renormalization. Our results generalize the seminal Cartier-Rota theory of classical Spitzer-type identities for commutative Rota-Baxter algebras. In the classical, commutative, case these identities can be understood as deriving from the theory of symmetric functions. Here we show that an analogous property holds for noncommutative Rota-Baxter algebras. That is, we show that functional identities in the noncommutative setting can be derived from the theory of noncommutative symmetric functions. Lie idempotents, and particularly the Dynkin idempotent, play a crucial role in the process. Their action on the pro-unipotent groups such as those of perturbative renormalization is described in detail along the way.

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Section: Lie Idempotents Actions On Pro-unipotent Groupsmentioning

confidence: 99%

A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion

Ebrahimi‐Fard¹,

Manchon²,

Patras³

2009

J. Noncommut. Geom.

103

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“…It has been shown that Qsym is a Hopf algebra [7,13] with the usual product of formal power series, counit that takes functions to their constant term, and the following coproduct,…”

Section: Quasisymmetric Functionsmentioning

confidence: 99%

“…In this paper we shall be concerned more with the Hopf algebraic structure of Qsym and its type B analog. One can carry the duality between Qsym and Sol := n≥0 Sol(A n ) to the level of Hopf algebras, see [13], but we will not do so here.…”

Section: Quasisymmetric Functionsmentioning

confidence: 99%

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Colored Posets and Colored Quasisymmetric Functions

Hsiao

Petersen

2010

Ann. Comb.

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Abstract. The colored quasisymmetric functions, like the classic quasisymmetric functions, are known to form a Hopf algebra with a natural peak subalgebra. We show how these algebras arise as the image of the algebra of colored posets. To effect this approach we introduce colored analogs of P -partitions and enriched P -partitions. We also frame our results in terms of Aguiar, Bergeron, and Sottile's theory of combinatorial Hopf algebras and its colored analog.

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“…Dans le cas Γ = 1, il est classique que la bigèbre (Q Y Γ , ·, ∆ ⋆ ) se ramèneà une bigèbre enveloppante d'algèbre de Lie libre (cf. [15]). La généralisation au cas Γ quelconque est facile et permet d'étendre Lià Q Y Γ , commeà la section précédente :…”

Section: Secondes Relations De Mélangeunclassified

Torseurs associés à certaines relations algébriques entre polyzêtas aux racines de l'unité

Racinet

2001

Comptes Rendus de l'Académie des Sciences - Series I - Mathemat

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version 3 -5 mars 2001 * Résumé. On décrit dans cette note une structure de torseur sur le schéma affine défini par une collection de relations Q-algébriques entre les polyzêtas aux racines de l'unité, i.e. les valeurs prises sur un groupe fixé de racines complexes de l'unité par les fonctions hyperlogarithmiquesà plusieurs variables. Dans le cas où la seule racine de l'unité est 1 (cas des polyzêtas), ces relations sont censéesêtre les seules entre ces nombres et le torseur obtenu devraitêtreégalà celui des associateurs de Drinfel'd. Les formules utilisées sont en général celles de l'action de Gal (Q/Q) sur la droite projective privée d'un nombre fini de points.Torsor structure associated to some algebraic relations between polyzetas at roots of unity Abstract. We describe in this note a torsor structure arising on the affine scheme defined by a system of Q-algebraic relations between polyzetas at roots of unity, i.e. values of hyperlogarithmic functions on a fixed finite subgroup Γ of C. For Γ = {1}, (polyzetas case), these relations are believed to span all the Q-algebraic relations between those numbers and this torsor should be equal to Drinfeld's, i.e. the Grothendieck-Teichmüller group acting on associators. The formulas for the general case are derived from the action of Gal (Q/Q) on the projective line minus finitely many points.

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Duality between Quasi-Symmetrical Functions and the Solomon Descent Algebra

Cited by 371 publications

References 0 publications

A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion

A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion

Colored Posets and Colored Quasisymmetric Functions

Torseurs associés à certaines relations algébriques entre polyzêtas aux racines de l'unité

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