Vol. 302, No. 2, 2019

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Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles

Yi Zhang, Dan Chen, Xing Gao and Yan-feng Luo

Vol. 302 (2019), No. 2, 741–766
Abstract

In this paper, we define a new coproduct on the space of decorated planar rooted forests to equip it with a weighted infinitesimal unitary bialgebraic structure. We introduce the concept of Ω-cocycle infinitesimal bialgebras of weight λ and then prove that the space of decorated planar rooted forests HRT(X,Ω), together with a set of grafting operations {Bω+ω Ω}, is the free Ω-cocycle infinitesimal unitary bialgebra of weight λ on a set X, involving a weighted version of a Hochschild 1-cocycle condition. As an application, we equip a free cocycle infinitesimal unitary bialgebraic structure on the undecorated planar rooted forests, which is the object studied in the well-known (noncommutative) Connes–Kreimer Hopf algebra.

Keywords
rooted forest, infinitesimal bialgebra, cocycle condition, operated algebra
Mathematical Subject Classification 2010
Primary: 16S10, 16T10, 16T30, 16W99, 17B60
Milestones
Received: 29 November 2018
Revised: 5 May 2019
Accepted: 10 May 2019
Published: 27 November 2019
Authors
Yi Zhang
School of Mathematics and Statistics
Lanzhou University
China
Dan Chen
School of Mathematics and Statistics
Lanzhou University
China
Xing Gao
School of Mathematics and Statistics
Key Laboratory of Applied Mathematics and Complex Systems
Lanzhou University
China
Yan-feng Luo
School of Mathematics and Statistics
Key Laboratory of Applied Mathematics and Complex Systems
Lanzhou University
China