Algebra and Applications 2. Группа авторов

       with:

      [1.114]

where stands for the k^th differential of fi, evaluated at x.

      PROOF.– We prove the result by induction on the number k of branches: for k = 1, we check:

      Now we can compute using the induction hypothesis and the fact that the vector fields are constant:

      COROLLARY 1.3 (closed formula).– With the notations of Corollary 1.2, for any rooted tree t with set of vertices and root r, each vertex v being decorated by a vector field , the vector field is given at x ∈ ℝⁿ by the following formula:

      [1.115]

      1.7.2. Novikov algebras

      A Novikov algebra is a right pre-Lie algebra which is also left NAP, namely, a vector space A together with a bilinear product ∗, such that, for any a, b, c ∈ A, we have:

      [1.116]

      [1.117]

      Novikov algebras first appeared in hydrodynamical equations (Balinskii and Novikov 1985; Osborn 1992). The prototype is a commutative associative algebra together with a derivation D, with the Novikov product being given by:

      [1.118]

      The free Novikov algebra on a set of generators has been given in Dzhumadil’daev and Löfwall (2002, section 7) in terms of some classes of rooted trees.

      1.7.3. Assosymmetric algebras

      An assosymmetric algebra is a vector space endowed with a bilinear operation that is both left and right pre-Lie, which means that the associator a * (b * c) – (a ∗ b) ∗ c is symmetric under the permutation group S₃. This notion was introduced by Kleinfeld (1957) (see also Hentzel et al. (1996)). All associative algebras are obviously assosymmetric; however, the converse is not true.

      1.7.4. Dendriform algebras

      A dendriform algebra (Loday 2001) over the field k is a k-vector space A endowed with two bilinear operations, denoted ≺ and ≻ and called right and left products, respectively, subject to the three axioms below:

      [1.119]

      [1.120]

      [1.121]

      We readily verify that these relations yield associativity for the product

      [1.122]

      However, at the same time-ordering, the dendriform relations imply that the bilinear product ⊳ defined by:

      [1.123]

is left pre-Lie. The associative operation * and the pre-Lie operation ⊳ define the same Lie bracket, and this is, of course, still true for the opposite (right) pre-Lie product ⊲:

      In the commutative case (commutative dendriform algebras are also named Zinbiel algebras (Loday 1995, 2001), the left and right operations are further required to identify, so that a ≻ b = b ≺ a. In this case, both pre-Lie products vanish. A natural example of a commutative dendriform algebra is given by the shuffle algebra in terms of half-shuffles (Schützenberger 1958/1959). Any associative algebra A equipped with a linear integral-like map I : A → A satisfying the integration by parts rule also gives a dendriform algebra, when a ≺ b := aI (b) and a ≻ b := I(a)b. The left pre-Lie product is then given by a ⊳ b = [I(a), b]. It is worth mentioning that Zinbiel algebras are also NAP algebras, as shown by the computation below (dating back to Schützenberger (1958/1959)):

      There also exists a twisted version of dendriform algebras, encompassing operators like the Jackson integral Iq (Ebrahimi-Fard and Manchon 2011). Returning to ordinary dendriform algebras, we observe that:

      [1.124]

      This identity generalizes to any number of elements, expressing the symmetrization of

      in terms of the associative product and the left pre-Lie product (Ebrahimi-Fard et al. 2008). For more on dendriform algebras and the associated pre-Lie structures, see Ebrahimi-Fard et al. (2008), Ebrahimi-Fard and Manchon (2009a, 2009b, 2011) and Ebrahimi-Fard’s note in the present volume.

      1.7.5. Post-Lie algebras

Post-Lie algebras have been introduced by Vallette (2007), independent of the introduction of the closely related notion of D-algebra in Munthe-Kaas and Wright (2008). A left post-Lie algebra on a field k is a k-vector space A together with a bilinear binary product ⊳ and a Lie bracket [–, –], such that

      for any a, b, c ∈ A. In particular, a post-Lie algebra A is a pre-Lie algebra if and only if the Lie bracket vanishes. The space of vector fields on a Lie group is a post-Lie algebra, and the free post-Lie algebra with one generator is the free Lie algebra on the linear span of planar rooted trees. The binary product ⊳ is given by left grafting, which is not pre-Lie anymore because of planarity. This is the starting point to the Lie-Butcher theory (see, for example, Lundervold and Munthe-Kaas (2013); Ebrahimi-Fard et al. (2015); Curry et al. (2019, 2020)).

1.8.