Noncommutative symmetric systems over associative algebras

Zhao, Wenhua

doi:10.1016/j.jpaa.2006.10.011

Cited by 4 publications

(1 citation statement)

References 35 publications

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“…In the reverse direction, Wenhua Zhao (see [33], [34], [35]) has for his part rediscovered the constructions of coarborification and then obtained in effect the mechanisms of arborification by dualizing and going to CK. Notably, Zhao's results concern in fact both plain and contracting arborification.…”

Section: Remarkmentioning

confidence: 99%

Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra

Fauvet¹,

Menous²

2017

Ann. Sci. École Norm. Sup.

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We give a natural and complete description of Ecalle's mould-comould formalism within a Hopf-algebraic framework. The arborification transform thus appears as a factorization of characters, involving the shuffle or quasishuffle Hopf algebras, thanks to a universal property satisfied by Connes-Kreimer Hopf algebra. We give a straightforward characterization of the fundamental process of homogeneous coarborification, using the explicit duality between decorated Connes-Kreimer and GrossmanLarson algebras. Finally, we introduce a new Hopf algebra that systematically underlies the calculations for the normalization of local dynamical systems.

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Section: Remarkmentioning

confidence: 99%

Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra

Fauvet¹,

Menous²

2017

Ann. Sci. École Norm. Sup.

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A noncommutative symmetric system over the Grossman-Larson Hopf algebra of labeled rooted trees

Zhao

2007

J Algebr Comb

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Abstract. In this paper, we construct explicitly a noncommutative symmetric (NCS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the NCS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a graded Hopf algebra homomorphism from the Connes-Kreimer Hopf algebra of labeled rooted forests to the Hopf algebra of quasi-symmetric functions. A connection of the coefficients of the third generating function of the constructed NCS system with the order polynomials of rooted trees is also given and proved.

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Differential operator specializations of noncommutative symmetric functions

Zhao

2007

Advances in Mathematics

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Let K be any unital commutative Q-algebra and z = (z 1 , ..., z n ) commutative or noncommutative free variables. Let t be a formal parameter which commutes with z and elements of K. We denote uniformly by K z and K[[t]] z the formal power series algebras of z over K and K[[t]], respectively. For any α ≥ 1, let D [α] z be the unital algebra generated by the differential operators of K z which increase the degree in z by at least α − 1 and Az , we introduce five sequences of differential operators of K z and show that their generating functions form a NCS (noncommutative symmetric) system ([Z4]) over the differential algebra D [α] z . Consequently, by the universal property of the NCS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [GKLLRT], we obtain a family of Hopf algebra homomorphisms S Ft :, which are also grading-preserving when F t satisfies certain conditions. Note that, the homomorphisms S Ft above can also be viewed as specializations of NCSFs by the differential operators of K z . Secondly, we show that, in both commutative and noncommutative cases, this family S Ft (with all n ≥ 1 and F t ∈ A[α] t z ) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions ([Ge], [MR], [S]) are also discussed.

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Noncommutative symmetric systems over associative algebras

Cited by 4 publications

References 35 publications

Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra

Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra

A noncommutative symmetric system over the Grossman-Larson Hopf algebra of labeled rooted trees

Differential operator specializations of noncommutative symmetric functions

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