Abstract
Combinatorics A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.
Highlights
The Hopf algebra of rooted forests first appeared in the work of A
Inspired by constructive quantum field theory [28], we propose in this article a noncommutative version of a Hopf algebra of graphs, by putting a total order on the set of edges
We prove that the vector space freely generated by these totally assigned graphs (TAGs) is a Hopf algebra
Summary
A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). We analyze the associated quadri-coalgebra and codendrifrom structures
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