Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1309.16023
DOI: 10.1142/S0218196714500283
author: Kurusch Ebrahimi-Fard
author: Alexander Lundervold
author: Dominique Manchon
published in: International Journal of Algebra and Computation
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1402.4761
Mathematics Subject Classification ID: 16T30
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 06A11
Mathematics Subject Classification ID: 05A18
zbMATH DE Number: 6356169
zbMATH Keywords: Dynkin Faà di Bruno Hopf algebras
zbMATH Keywords: Bell polynomials
zbMATH Keywords: partitions
zbMATH Keywords: quasideterminants
zbMATH Keywords: Faà di Bruno formula
zbMATH Keywords: incidence Hopf algebras
MaRDI profile type: Publication
arXiv ID: 1402.4761
cites work: Exponential polynomials
cites work: Non-commutative Hopf algebra of formal diffeomorphisms.
cites work: On posets and Hopf algebras
cites work: COMBINATORIAL HOPF ALGEBRAS IN QUANTUM FIELD THEORY I
cites work: Quasideterminants
cites work: Noncommutative symmetric functions
cites work: Determinants of matrices over noncommutative rings
cites work: A \(q\)-analog of the exponential formula
cites work: Incidence algebra antipodes and Lagrange inversion in one and several variables
cites work: A \(q\)-analogue of Faà di Bruno's formula
cites work: Some applications of the \(q\)-exponential formula
cites work: The Curious History of Faa di Bruno's Formula
cites work: Coalgebras and Bialgebras in Combinatorics
cites work: Lie-Butcher theory for Runge-Kutta methods
cites work: Noncommutative symmetric functions and Lagrange inversion
cites work: Derivatives of composite functions
cites work: Q3243274
cites work: On the foundations of combinatorial theory I. Theory of M�bius Functions
cites work: NONCOMMUTATIVE BELL POLYNOMIALS
cites work: Differential polynomial expressions related to the Kadomtsev–Petviashvili and Korteweg–de Vries hierarchies
cites work: Antipodes and incidence coalgebras
cites work: Incidence Hopf algebras
cites work: Q4889899
cites work: Free Hopf algebras generated by coalgebras
OpenAlex ID: W2153028405
title: Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras.
review text: Bell polynomials appear in the combinatorics of set partitions and in the composition of formal diffeomorphisms of \(\mathbb R\), or equivalently in the coproduct of the Faà di Bruno Hopf algebra. They admit a recursive definition, and explicit descriptions in terms of rooted trees or determinants. These results are here extended to a noncommutative version of these polynomials: they can be inductively defined, admit an explicit description in terms of planar rooted trees or quasideterminants. They are related to the composition of diffeomorphisms on varieties, or in a similar way appear in the coproduct of the noncommutative Dynkin Faà di Bruno Hopf algebra.NEWLINENEWLINE The language of incidence Hopf algebras is used in this text for the Faà di Bruno and Dynkin Faà di Bruno Hopf algebras, and a new description of the antipode of such objects is given by a quasideterminant. The text also contains a discussion on Möbius inversion in certain variants of Faà di Bruno and Dynkin Faà di Bruno Hopf algebras built from Bell polynomials.
reviewed by: Loïc Foissy
description: scientific article; zbMATH DE number 6356169