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By Loday J.-L., Vallette B.

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This construction is functorial both in C and in A. Let g : A → A be a morphism of dga algebras and f : C → C be a morphism of dga coalgebras. Consider C ⊗α A and C ⊗α A two twisted tensor products. We say that the morphisms f and g are compatible with the twisting morphisms α and α if α ◦ f = g ◦ α. One can show that f ⊗ g : C ⊗α A → C ⊗α A is then a morphism of chain complexes. In the weight-graded context, we require that the twisting morphisms preserve the weight. In this case, the following lemma states that if two among these three morphisms are quasi-isomorphisms, then so is the third one.

The jacobiator is the sum of 12 elements that can be grouped into 3 × 4. The pre-Lie relators for (x, y, z), (y, z, x), (z, x, y) account for these 3 packets. 3. Remark. The above Lemma is valid for both the right pre-Lie product and the left pre-Lie product. Observe that for left pre-Lie algebras the linear map L(x) defined by L(x)(y) := {x, y} satisfies the following functional equation: L([x, y]) = [L(x), L(y)]. 4. Examples. (a) An associative algebra is an example of a pre-Lie algebra since the associator is trivial.

We denote the set of (p, q)-unshuffles by Sh−1 p,q . 3. For any n = p + q and σ ∈ Sn there exist unique permutations α ∈ Sp , β ∈ Sq and ω ∈ Sh(p, q) such that: σ = ω · (α × β). Proof. The permutation α is the unique element of Aut{1, . . , p} such that σ(α−1 (i)) < σ(α−1 (i + 1)) for any i = 1, . . , p − 1. The permutation β is the unique element of Aut{p + 1, . . , p + q} such that σ(β −1 (i)) < σ(β −1 (i + 1)) for any i = p + 1, . . , p + q − 1. Since ω := σ · (α × β)−1 is a (p, q)-shuffle, we are done.

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Algebraic operads by Loday J.-L., Vallette B.


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