Algebra and Applications 2

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is symmetric in X and Y , due to the fact that the two constant vector fields XO and YO commute. The second assertion is left as an exercise for the reader. □

With the notations of section 1.6.4, there is a unique NAP algebra morphism

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the frozen Cayley map , such that . By also considering the unique NAP algebra morphism , the maps GX,O ( t ) : ℝ n→ ℝ nare called the frozen elementary differentials .

PROPOSITION 1.18.– For any rooted tree t, each vertex v being decorated by a vector field Xv, the vector field is given at x ∈ ℝ n by the following recursive procedure: if the decorated tree t is obtained by grafing all of its branches tk on the root r decorated by the vector field , that is, if it writes , then :

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with:

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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 ∈ ℝ n by the following formula:

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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:

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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:

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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.

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 S 3. 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.

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:

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We readily verify that these relations yield associativity for the product

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

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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)):