Algebra and Applications 2. Группа авторов

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      where the product runs over the incoming vertices of v.

      Now fix a vector field X on ℝn and consider the map dX from undecorated rooted trees to vector field-decorated rooted trees, which decorates each vertex by X. It is obviously a pre-Lie algebra morphism, and image is the unique pre-Lie algebra morphism that sends the one-vertex tree • to the vector field X.

      1.6.5. B-series, composition and substitution

      B-series have been defined by Hairer and Wanner, following the pioneering work of Butcher (1963) on Runge-Kutta methods for the numerical resolution of differential equations. Remarkably enough, rooted trees revealed to be an adequate tool not only for vector fields, but also for the numerical approximation of their integral curves. Butcher discovered that the Runge-Kutta methods formed a group (since then called the Butcher group), which was nothing but the character group of the Connes-Kreimer Hopf algebra ℋCK (Brouder 2000).

      [1.100]image

      where α is any linear form on image (here, σ(s) is the symmetry factor of the tree, that is, the order of its group of automorphisms). It matches the usual notion of B-series (Hairer et al. 2002) when A is the pre-Lie algebra of vector fields on ℝn (it is also convenient to set Fa(∅) = 1). In this case, the vector fields Fa(t) for a tree t are differentiable maps from ℝn to ℝn called elementary differentials. B-series can be composed coefficient wise, as series in the indeterminate h, whose coefficients are maps from ℝn to ℝn. The same definition with trees decorated by a set of colors image leads to straightforward generalizations. For example, the P-series used in partitioned Runge-Kutta methods (Hairer et al. 2002) correspond to bi-coloured trees.

      A slightly different way of defining B-series is the following: consider the unique pre-Lie algebra morphism

with respect to the grading. We further extend it to the empty tree by setting ∙a(∅) = 1. We then have:

      [1.101]image

      where image is the isomorphism from image to image given by the normalized dual basis (see section 1.6.3).

      We restrict ourselves to B-series B(α; a) with α(∅) = 1. Such αs are in one-to-one correspondence with characters of the algebra of forests (which is the underlying algebra of ℋCK) by setting:

      [1.102]image

      The Hairer-Wanner theorem (Hairer et al. 2002, Theorem III.1.10) says that the composition of B-series corresponds to the convolution product of characters of ℋCK, namely:

      [1.103]image

      where linear forms α, β on image and their character counterparts are identified modulo the above correspondence.

      PROPOSITION 1.16.– For any linear forms α,β on with α(• = 1), we have:

      [1.104]image

      where α is the multiplicatively extended to forests, β is seen as an infinitesimal character ofCK, and * is the dualization of the left coaction Φ ofonCK defined in section 1.6.3.

      The condition α(•) = 1 is in fact dropped in Calaque et al. (2011, Proposition 15): the price to pay is that one has to replace the Hopf algebra ℋ by a non-connected bialgebra image with a suitable coproduct, such that ℋ is obtained as the quotient image, where image is the ideal generated by • – 1. The substitution product * then coincides with the one considered in Chartier et al. (2010) via natural identifications.

      1.7.1. NAP algebras

      NAP algebras (NAP for Non-Associative Permutative) appear under this name in Livernet (2006), and under the name “left- (right-)commutative algebras” in Dzhumadil’daev and Löfwall (2002). They can be seen in some sense as a “simplified version” of pre-Lie algebras. Saidi showed that the pre-Lie operad is a deformation of the NAP operad in a precise sense, involving the notion of current-preserving operad (Saidi 2014).

      1.7.1.1. Definition and general properties

      A

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