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The Connes-Kreimer Hopf Algebra

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A Combinatorial Perspective on Quantum Field Theory

Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 15))

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Abstract

A coproduct takes an object and returns a sum of ways to break the object into two pieces. We get a combinatorial Hopf algebra if the product and coproduct are compatible in a specific way.

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Notes

  1. 1.

    A bridge is an edge which upon removal increases the number of connected components of a graph.

  2. 2.

    Personal communication with Spencer Bloch and Dirk Kreimer.

  3. 3.

    Some more details on the connection between Feynman rules coming from the universal property and the exponential map can be found in lecture notes of Erik Panzer http://people.math.sfu.ca/~kyeats/seminars/Panzer 0- 02.pdf; my understanding of the connection between the convolution property and the exponential map is based on personal communication with Jason Bell and Julian Rosen.

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada

    Karen Yeats

  2. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, Canada

    Karen Yeats

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Yeats, K. (2017). The Connes-Kreimer Hopf Algebra. In: A Combinatorial Perspective on Quantum Field Theory. SpringerBriefs in Mathematical Physics, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-47551-6_4

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