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

DOI: 10.2140/pjm.2019.302.741

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