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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)
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