#### 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 $\Omega$-cocycle infinitesimal bialgebras of weight $\lambda$ and then prove that the space of decorated planar rooted forests ${H}_{RT}\left(X,\Omega \right)$, together with a set of grafting operations $\left\{{B}_{\omega }^{+}\mid \omega \in \Omega \right\}$, is the free $\Omega$-cocycle infinitesimal unitary bialgebra of weight $\lambda$ 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