Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles

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 $H_{\mathrm{RT}}(X,\Omega )$, together with a set of grafting operations $\{B_{\omega }^{+}\mid \omega \in \Omega \}$, 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. Finally, we construct a new pre-Lie algebraic structure on decorated planar rooted forests.

arXiv comment20 pages. arXiv admin note: substantial text overlap with arXiv:1810.10790

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Cited by in corpus (2)

• Cohomology and deformations of weighted Rota-Baxter operators
• Hopf algebra of multi-decorated rooted forests, free matching Rota-Baxter algebras and Gröbner-Shirshov bases