We define two coproducts for cycle-free oriented graphs, thus building up two commutative connected graded Hopf algebras, such that one is a comodule-coalgebra on the other, thus generalizing the result obtained in [2] for Hopf algebras of rooted trees.
@article{CML_2012__4_1_A4_0,
author = {Dominique Manchon},
title = {On bialgebras and {Hopf} algebras of oriented graphs},
journal = {Confluentes Mathematici},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {4},
number = {1},
year = {2012},
doi = {10.1142/S1793744212400038},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744212400038/}
}
TY - JOUR AU - Dominique Manchon TI - On bialgebras and Hopf algebras of oriented graphs JO - Confluentes Mathematici PY - 2012 VL - 4 IS - 1 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744212400038/ DO - 10.1142/S1793744212400038 LA - en ID - CML_2012__4_1_A4_0 ER -
%0 Journal Article %A Dominique Manchon %T On bialgebras and Hopf algebras of oriented graphs %J Confluentes Mathematici %D 2012 %V 4 %N 1 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744212400038/ %R 10.1142/S1793744212400038 %G en %F CML_2012__4_1_A4_0
Dominique Manchon. On bialgebras and Hopf algebras of oriented graphs. Confluentes Mathematici, Tome 4 (2012) no. 1. doi : 10.1142/S1793744212400038. https://cml.centre-mersenne.org/articles/10.1142/S1793744212400038/
[1] Ch. Brouder, Runge–Kutta methods and renormalization, Euro. Phys. J. C 12 (2000) 512–534.
[2] D. Calaque, K. Ebrahimi-Fard and D. Manchon, Two interacting Hopf algebras of trees, Adv. Appl. Math. 47 (2011) 282–308.
[3] Ph. Chartier, E. Hairer and G. Vilmart, Numerical integrators based on modified differential equations, Math. Comp. 76 (2007) 1941–1953.
[4] A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199 (1998) 203–242.
[5] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann–Hilbert problem I: The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys. 210 (2000) 249–273.
[6] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann–Hilbert problem. II. The β-function, diffeomorphisms and the renormal- ization group, Commun. Math. Phys. 216 (2001) 215–241.
[7] L. Foissy, Les algèbres de Hopf des arbres enracinés décorés I, II, Bull. Sci. Math. 126 (2002) 193–239, 249–288.
[8] D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 303–334.
[9] J.-L. Loday and M. Ronco, Combinatorial Hopf algebras, in Quantum of Maths, Clay Math. Proc., Vol. 11 (Amer. Math. Soc., 2010), pp. 347–383.
[10] D. Manchon and A. Sa¨ıdi, Lois pré-Lie en interaction, Comm. Alg. 39 (2011) 3662– 3680.
[11] A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found. Comput. Math. 6 (2006) 387–426.
Cité par Sources :