Combinatorial Hopf algebras from renormalization
Christian Brouder
, Alessandra Frabetti
and Frédéric Menous
DOI: 10.1007/s10801-010-0227-7
Abstract
In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Faà di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field.
We also describe two general ways to define the associative product in such Hopf algebras, the first one by recursion, and the second one by grafting and shuffling some decorated rooted trees.
Pages: 557–578
Keywords: keywords Hopf algebra; combinatorial; brace; tree; renormalization
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References
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2. Brouder, Ch., Frabetti, A.: QED Hopf algebras on planar binary trees. J. Algorithms 267, 298-322 (2003)
3. Brouder, Ch., Frabetti, A., Krattenthaler, Ch.: Non-commutative Hopf algebra of formal diffeomorphisms. Adv. Math. 200, 479-524 (2006)
4. Brouder, Ch., Schmitt, W.: Renormalization as a functor on bialgebras. J. Pure Appl. Algebra 209, 477-495 (2007)
5. Chapoton, F.: Un théorème de Cartier-Milnor-Moore-Quillen pour les bigèbres dendriformes et les algèbres brace. J. Pure Appl. Algebra 168(1), 1-18 (2002)
6. Foissy, L.: Les algèbres de Hopf des arbres enracinés décorés, II. Bull. Sci. Math. 126, 249-288 (2002)
7. Grossman, R., Larson, R.G.: Hopf-algebraic structure of families of trees. J. Algebra 126, 184-210 (1989)
8. Guin, D., Oudom, J.-M.: On the Lie envelopping algebra of a pre-Lie algebra. K-Theory 2(1), 147- 167 (2008) J Algebr Comb (2010) 32: 557-578
9. Kreimer, D.: On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math.
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