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Feynman Graphs, Rooted Trees, and Ringel-Hall Algebras

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Abstract

We construct symmetric monoidal categories \({\mathcal{LRF}, \mathcal{LFG}}\) of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf algebras of \({\mathcal{LRF}, \mathcal{LFG}}\), \({{\bf H}_\mathcal{LRF}, {\bf H}_\mathcal{LFG}}\) are dual to the corresponding Connes-Kreimer Hopf algebras on rooted trees and Feynman diagrams. We thus obtain an interpretation of the Connes-Kreimer Lie algebras on rooted trees and Feynman graphs as Ringel-Hall Lie algebras.

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Authors and Affiliations

  1. Department of Mathematics, University of Chicago, Chicago, IL, USA

    Kobi Kremnizer

  2. Department of Mathematics, Boston University, Boston, MA, USA

    Matt Szczesny

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  1. Kobi Kremnizer
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  2. Matt Szczesny
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Correspondence to Matt Szczesny.

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Communicated by A. Connes

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Kremnizer, K., Szczesny, M. Feynman Graphs, Rooted Trees, and Ringel-Hall Algebras. Commun. Math. Phys. 289, 561–577 (2009). https://doi.org/10.1007/s00220-008-0694-z

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  • DOI: https://doi.org/10.1007/s00220-008-0694-z

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