#### 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