By Jean-Louis Loday, Bruno Vallette

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The integer r is called the degree of f and denoted by |f |. So we have |f (v)| = |f | + |v|. If the non-negative components are all zero, then it is helpful to write V n := V−n . We will say that V• = n Vn is homologically graded, and that V • = n V n is cohomologically graded. There are obvious notions of subvector space and quotient vector space in the graded framework. The grading of the tensor product V ⊗ W of two graded spaces V and W is described explicitly as: (V ⊗ W )n := Vi ⊗ Wj . i+j=n Hence V ⊗n is graded and so is the tensor module T (V ).

1. Pre-Lie algebra. By definition a (right) pre-Lie algebra is a vector space A equipped with a binary operation {x, y} which satisfies the following relation, called pre-Lie relation: {{x, y}, z} − {x, {y, z}} = {{x, z}, y} − {x, {z, y}} . In plain words, the associator (left side part of the equality) is right-symmetric. e. x, y := {y, x}) the associator is left-symmetric. It appeared in the work of Gerstenhaber [Ger63] and Vinberg [Vin63] in differential geometry and in several other papers subsequently.

Ik )-shuffle and the set Sh(i1 , . . , ik ) are defined analogously. Following Jim Stasheff, we call unshuffle the inverse of a shuffle. For instance the three (1, 2)-unshuffles are [1 2 3], [2 1 3] and [3 1 2]. 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, .