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Infinitesimal unitary Hopf algebras and planar rooted forests

Authors
  • Gao, Xing1
  • Wang, Xiaomeng2
  • 1 Lanzhou University, School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou, Gansu, 730000, People’s Republic of China , Lanzhou (China)
  • 2 Lanzhou University, School of Mathematics and Statistics, Lanzhou, Gansu, 730000, People’s Republic of China , Lanzhou (China)
Type
Published Article
Journal
Journal of Algebraic Combinatorics
Publisher
Springer US
Publication Date
Jun 27, 2018
Volume
49
Issue
4
Pages
437–460
Identifiers
DOI: 10.1007/s10801-018-0830-6
Source
Springer Nature
Keywords
License
Yellow

Abstract

Infinitesimal bialgebras were introduced by Joni and Rota. An infinitesimal bialgebra is at the same time an algebra and coalgebra, in such a way that the comultiplication is a derivation. Twenty years after Joni and Rota, Aguiar introduced the concept of an infinitesimal (non-unitary) Hopf algebra. In this paper, we study infinitesimal unitary bialgebras and infinitesimal unitary Hopf algebras, in contrary to Aguiar’s approach. Using an infinitesimal version of the Hochschild 1-cocycle condition, we prove, respectively, that a class of decorated planar rooted forests is the free cocycle infinitesimal unitary bialgebra and free cocycle infinitesimal unitary Hopf algebra on a set. As an application, we obtain that the planar rooted forests are the free cocycle infinitesimal unitary Hopf algebra on the empty set.

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