Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras
type: Item
instance of: scholarly article
title: Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras
zbMATH Open document ID: 1476.16041
author: Yi Zhang
author: Xing Gao
author: Yanfeng Luo
published in: Journal of Algebraic Combinatorics
publication date: 2021-07-07T00:00:00Z
review text: A weighted infinitesimal unitary bialgebra is a module \(A\), which is simultaneously an algebra (possibly without a unit) and a coalgebra (possibly without a counit) such that the coproduct \(\Delta\) is a weighted derivation of \(A\): \[ \Delta(ab)=a\Delta(b)+\Delta(a)b+\lambda a\otimes b,\quad \lambda\in k \] Take an associative classical Young-Baxter equation \[ r_{13}r_{12}-r_{12}r_{23}+r_{23}r_{13}=\lambda r_{13} \] Then any solution of this equarion in an algebra \(A\) endows \(A\) with an infinitesimal unitary bialgebra of weight zero. If one takes an nonhomogenous associative classical Yang-Baxter equation \[ r_{13}r_{12}-r_{12}r_{23}+r_{23}r_{13}=\lambda r_{13} \] then the coproduct \[ \Delta_r(a)=a\dot r-r\dot a-\lambda (a\otimes 1) \] is weighter derivation [\textit{O. Ogievetsky} and \textit{T. Popov}, Adv. Theor. Math. Phys. 14, No. 2, 439--505 (2010; Zbl 1208.81118)]. Also such algebras appear naturally in combinatorics [\textit{S. A. Joni} and \textit{G. C. Rota}, Stud. Appl. Math. 61, 93--139 (1979; Zbl 0471.05020)]. The authors ``introduce the concept of symmetric \(1\)-cocycle condition, which is derived from the dual of the Hochschild cohomology. Also they study the universal properties of the space of decorated planar rooted forests with the symmetric \(1\)-cocycle, leading to the notation of a weighted \(\Omega\)-cocycle infinitesimal unitary bialgebra. As an application, they obtain the initial object in the category of free cocycle infinitesimal unitary bialgebras on the undecorated planar rooted forests, which is the object studied in the well-known noncommutative Connes-Kreimer Hopf algebra. Finally, they construct a pre-Lie algebra on decorated planar rooted forests''.
Mathematics Subject Classification ID: 16W99
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16T10
Mathematics Subject Classification ID: 16T30
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 7369363
zbMATH Keywords: rooted forest
zbMATH Keywords: infinitesimal bialgebra
zbMATH Keywords: cocycle condition
zbMATH Keywords: pre-Lie algebra
reviewed by: Dmitry Artamonov
MaRDI profile type: Publication
full work available at URL: s10801-020-00942-7
OpenAlex ID: W3084185036
cites work: Infinitesimal Hopf algebras
cites work: On the associative analog of Lie bialgebras
cites work: Infinitesimal Hopf algebras and the cd-index of polytopes
cites work: Q4466631
cites work: Gröbner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras.
cites work: Combinatorial Hopf algebras from renormalization
cites work: Algebraic renormalisation of regularity structures
cites work: Renormalization in quantum field theory and the Riemann-Hilbert problem. I: The Hopf algebra structure of graphs and the main theorem
cites work: Hopf algebras, renormalization and noncommutative geometry
cites work: Q3819193
cites work: Finite dimensional comodules over the Hopf algebra of rooted trees
cites work: The Hopf algebras of decorated rooted trees. I
cites work: The Hopf algebras of decorated rooted trees. II
cites work: The infinitesimal Hopf algebra and the poset of planar forests.
cites work: The Infinitesimal Hopf Algebra and the Operads of Planar Forests
cites work: Classification of systems of Dyson-Schwinger equations in the Hopf algebra of decorated rooted trees.
cites work: The cohomology structure of an associative ring
cites work: Rota's classification problem, rewriting systems and Gröbner-Shirshov bases
cites work: Hopf-algebraic structure of families of trees
cites work: Q3166202
cites work: Operated semigroups, Motzkin paths and rooted trees
cites work: Renormalization by Birkhoff-Hopf factorization and by generalized evaluators: a case study
cites work: Differential type operators and Gröbner-Shirshov bases.
cites work: Groups with Multiple Operators
cites work: Combinatorics of rooted trees and Hopf algebras
cites work: Comparison of Hopf algebras on trees.
cites work: Coalgebras and Bialgebras in Combinatorics
cites work: On the Hopf algebra strucutre of perturbative quantum field theories
cites work: Renormalization and Mellin Transforms
cites work: Q3275000
cites work: Q5288307
cites work: Hopf algebra of the planar binary trees
cites work: On the structure of cofree Hopf algebras
cites work: Q3118326
cites work: On the Connes-Kreimer construction of Hopf algebras
cites work: \(R\)-matrices in rime
cites work: Left counital Hopf algebras on bi-decorated planar rooted forests and Rota-Baxter systems
cites work: Q5513583
cites work: Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras
cites work: Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles
cites work: Weighted infinitesimal unitary bialgebras, pre-Lie, matrix algebras, and polynomial algebras
DOI: 10.1007/S10801-020-00942-7
Recommended article: Weighted infinitesimal unitary bialgebras on free monoid algebras
Recommended article: Drinfeld Double for Braided Infinitesimal Hopf Algebras
Recommended article: Infinitesimal Hopf algebras
Recommended article: Rota-Baxter systems, dendriform algebras and covariant bialgebras
Recommended article: Analogs of \(q\)-Serre relations in the Yang-Baxter algebras
Recommended article: Q4656929
Recommended article: A new approach to antisymmetric infinitesimal bialgebras
Recommended article: -Rota–Baxter Operators, Infinitesimal Hom-bialgebras and the Associative (Bi)Hom-Yang–Baxter Equation
Recommended article: Q3351552
Recommended article: On the associative analog of Lie bialgebras
description: scientific article; zbMATH DE number 7369363