# zbMATH — the first resource for mathematics

Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series. (English) Zbl 1235.16032
The well-known Connes-Kreimer Hopf algebra $$\mathcal H_{CK}$$ is the commutative algebra generated by rooted trees. Its coproduct is given by admissible cuts on a rooted tree, and it is graded by the number of vertices.
In the paper under review, the authors introduce a commutative Hopf algebra $$\mathcal H$$ of rooted forests, i.e., products of a finite number of rooted trees, whose connected components contain at least one edge. It is graded by the number of edges. A subforest $$s$$ of a tree $$t$$ is a collection of pairwise disjoint subtrees, and $$t/s$$ is the tree obtained by contracting each connected component of $$s$$ onto a vertex. The coproduct of $$t$$ is the sum over all subforests $$s$$ of $$t$$ of $$s\otimes t/s$$. The antipode of $$\mathcal H$$ is defined recursively, but an explicit formula is also given using partitions of a tree into subforests. $$\mathcal H_{CK}$$ is an $$\mathcal H$$-bicomodule, and the authors determine a relation of this $$\mathcal H$$-bicomodule structure with the coproduct on $$\mathcal H_{CK}$$.
Using a pre-Lie product on the primitive part of the graded dual of $$\mathcal H$$, the authors define a B-series involving elementary differentials, which is the usual B-series of E. Hairer, C. Lubich and G. Wanner [Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. Berlin: Springer (2002; Zbl 0994.65135)]. This enables them to recover recent results in the field of numerical methods for differential equations due to P. Chartier, E. Hairer and G. Vilmart [Found. Comput. Math. 10, No. 4, 407-427 (2010; Zbl 1201.65124)] and to A. Murua [Found. Comput. Math. 6, No. 4, 387-426 (2006; Zbl 1116.17004)].

##### MSC:
 16T30 Connections of Hopf algebras with combinatorics 05C05 Trees 05E15 Combinatorial aspects of groups and algebras (MSC2010)
Full Text:
##### References:
 [1] Abe, E., Hopf algebras, (1980), Cambridge University Press Cambridge · Zbl 0476.16008 [2] Agrachev, A.; Gamkrelidze, R., Chronological algebras and nonstationary vector fields, J. sov. math., 17, 1650-1675, (1981) · Zbl 0473.58021 [3] Bergman, G.M., Everybody knows what a Hopf algebra is, (), 25-48 · Zbl 0569.16005 [4] Brouder, Ch., Runge-Kutta methods and renormalization, Eur. phys. J. C part. fields, 12, 512-534, (2000) [5] Butcher, J.C., An algebraic theory of integration methods, Math. comp., 26, 79-106, (1972) · Zbl 0258.65070 [6] Butcher, J.C., The numerical analysis of ordinary differential equations. Runge-Kutta and general linear methods, (2008), Wiley Chichester · Zbl 0616.65072 [7] Cayley, A., A theorem on trees, Q. J. math., 23, 376-378, (1889) · JFM 21.0687.01 [8] Chapoton, F., Rooted trees and an exponential-like series, preprint · Zbl 1302.05205 [9] Chapoton, F.; Livernet, M., Pre-Lie algebras and the rooted trees operad, Int. math. res. not., 2001, 395-408, (2001) · Zbl 1053.17001 [10] Chapoton, F.; Livernet, M., Relating two Hopf algebras built from an operad, Int. math. res. not., (2007), Art. ID rnm131, 27 pp · Zbl 1144.18006 [11] Chartier, P.; Hairer, E.; Vilmart, G., Algebraic structures of B-series, Found. comput. math., 10, 4, 407-427, (2010) · Zbl 1201.65124 [12] Chartier, Ph.; Hairer, E.; Vilmart, G., Numerical integrators based on modified differential equations, Math. comp., 76, 1941-1953, (2007) · Zbl 1122.65059 [13] Ph. Chartier, E. Hairer, G. Vilmart, Algebraic structures of B-series, preprint, 2009. · Zbl 1201.65124 [14] Chartier, Ph.; Murua, A., An algebraic theory of order, M2AN math. model. numer. anal., 43, 607-630, (2009) · Zbl 1229.05302 [15] Connes, A.; Kreimer, D., Hopf algebras, renormalization and noncommutative geometry, Comm. math. phys., 199, 203-242, (1998) · Zbl 0932.16038 [16] Connes, A.; Kreimer, D., Renormalization in quantum field theory and the Riemann-Hilbert problem I. the Hopf algebra structure of graphs and the main theorem, Comm. math. phys., 210, 249-273, (2000) · Zbl 1032.81026 [17] Dzhumadilʼdaev, A.; Löfwall, C., Trees, free right-symmetric algebras, free Novikov algebras and identities, Homology, homotopy appl., 4, 165-190, (2002) · Zbl 1029.17001 [18] Ebrahimi-Fard, K.; Manchon, D., A Magnus- and fer-type formula in dendriform algebras, Found. comput. math., 9, 295-316, (2009) · Zbl 1173.17002 [19] Ebrahimi-Fard, K.; Manchon, D., Dendriform equations, J. algebra, 322, 4053-4079, (2009) · Zbl 1229.17001 [20] Gelfand, I.M.; Krob, D.; Lascoux, A.; Leclerc, B.; Retakh, V.; Thibon, J.-Y., Noncommutative symmetric functions, Adv. math., 112, 218-348, (1995) · Zbl 0831.05063 [21] Foissy, L., LES algèbres de Hopf des arbres enracinés décorés I + II, Bull. sci. math., Bull. sci. math., 126, 249-288, (2002), thèse, Univ. de Reims, 2002 · Zbl 1013.16027 [22] Grossman, R.; Larson, R.G., Hopf-algebraic structure of families of trees, J. algebra, 126, 184-210, (1989) · Zbl 0717.16029 [23] Hairer, E.; Lubich, C.; Wanner, G., Geometric numerical integration structure-preserving algorithms for ordinary differential equations, Springer ser. comput. math., vol. 31, (2002), Springer-Verlag Berlin · Zbl 0994.65135 [24] Hoffman, M., Combinatorics of rooted trees and Hopf algebras, Trans. amer. math. soc., 355, 3795-3811, (2003) · Zbl 1048.16023 [25] Hoffman, M., Quasi-shuffle products, J. algebraic combin., 11, 49-68, (2000) · Zbl 0959.16021 [26] Joni, S.A.; Rota, G.-C., Coalgebras and bialgebras in combinatorics, Stud. appl. math., 61, 93-139, (1979) · Zbl 0471.05020 [27] Kreimer, D., Chenʼs iterated integral represents the operator product expansion, Adv. theor. math. phys., 3, 627-670, (1999) · Zbl 0971.81093 [28] Kreimer, D., The combinatorics of (perturbative) quantum field theories, Phys. rep., 363, 387-424, (2002) · Zbl 0994.81080 [29] Livernet, M., A rigidity theorem for pre-Lie algebras, J. pure appl. algebra, 207, 1-18, (2006) · Zbl 1134.17001 [30] Loday, J.-L.; Ronco, M., Hopf algebra of the planar binary trees, Adv. math., 139, 293-309, (1998) · Zbl 0926.16032 [31] Magnus, W., On the exponential solution of differential equations for a linear operator, Comm. pure appl. math., 7, 649-673, (1954) · Zbl 0056.34102 [32] Manchon, D., Hopf algebras and renormalisation, (), 365-427 · Zbl 1215.81071 [33] Manchon, D.; Saïdi, A., Lois pré-Lie en interaction, (2008), Comm. Algebra, in press, preprint [34] Munthe-Kaas, H.; Wright, W., On the Hopf algebraic structure of Lie group integrators, Found. comput. math., 8, 227-257, (2008) · Zbl 1147.16028 [35] Murua, A., The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found. comput. math., 6, 387-426, (2006) · Zbl 1116.17004 [36] Panaite, F., Relating the Connes-kreimer and the grossman-larson Hopf algebras built on rooted trees, Lett. math. phys., 51, 211-219, (2000) · Zbl 0959.16023 [37] Schmitt, W., Incidence Hopf algebras, J. pure appl. algebra, 96, 299-330, (1994) · Zbl 0808.05101 [38] Sweedler, M.E., Hopf algebras, (1969), Benjamin New York · Zbl 0194.32901 [39] G. Vilmart, Étude dʼintégrateurs géométriques pour des équations différentielles, PhD thesis, Univ. Rennes I (INRIA Rennes) and Univ. Genève (Section de mathématiques), available at http://sma.epfl.ch/ vilmart/. [40] Zhao, W., A noncommutative symmetric system over the grossman-larson Hopf algebra of labeled rooted trees, J. algebraic combin., 28, 235-260, (2008) · Zbl 1152.05057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.