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Abstract
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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
-cocycle infinitesimal
bialgebras of weight
and then prove that the space of decorated planar rooted forests
, together with a set of
grafting operations
, is the
free
-cocycle infinitesimal
unitary bialgebra of weight
on a set
,
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.
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Keywords
rooted forest, infinitesimal bialgebra, cocycle condition,
operated algebra
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Mathematical Subject Classification 2010
Primary: 16S10, 16T10, 16T30, 16W99, 17B60
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Milestones
Received: 29 November 2018
Revised: 5 May 2019
Accepted: 10 May 2019
Published: 27 November 2019
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