Algebra and Applications 2. Группа авторов

      is clearly a morphism of left -modules. It is immediately seen by induction that for any a₁,…,an ∈ A, we have η(a₁ … an) = a₁ … an + v, where v is a sum of terms of degree < n – 1. This proves the theorem. □

      REMARK 1.3.– Let us recall that the symmetrization map , uniquely determined by σ(an) = an for any a ∈ A and any integer n, is an isomorphism for the two A_Lie-module structures given by the adjoint action. This is not the case for the map η defined above. The fact that it is possible to replace the adjoint action of on itself by the simple left multiplication is a remarkable property ofpre-Lie algebras, and makes Theorem 1.3 different from the usual Lie algebra PBW theorem.

      Let us finally note that, if p stands for the projection from S(A) onto A, for any a₁,…, ak ∈ A, we easily get:

[1.54]

      by a simple induction on k. The linear isomorphism η transfers the product of the enveloping algebra into a noncommutative product * on defined by:

      [1.55]

Suppose now that A is endowed with a complete decreasing compatible filtration as in section 1.4.2. This filtration induces a complete decreasing filtration S(A) = S(A)₀ ⊃ S(A)₁ ⊃ S(A)₂ ⊃ …, and the product * readily extends to the completion . For any a ∈ A, the application of equation [1.54] gives:

      [1.56]

      as an equality in the completed symmetric algebra .

According to equation [1.48], we can identify the pro-unipotent group {e^*a, a ∈ A} ⊂ and the group of formal flows of the pre-Lie algebra A by means of the projection p, namely:

      [1.57]

      for any a, b ∈ A.

1.5. Algebraic operads

      An operad is a combinatorial device which appeared in algebraic topology (May 1972), coined for coding “types of algebras”. Hence, for example, a Lie algebra is an algebra over some operad denoted by LIE, an associative algebra is an algebra over some operad denoted by ASSOC, a commutative algebra is an algebra over some operad denoted by COM and so on.

      1.5.1. Manipulating algebraic operations

      Algebra starts, in most cases, with some set E and some binary operation * : E × E → E. The set E shows some extra structure most of the time. Here, we will stick to the linear setting, where E is replaced by a vector space V (over some base field k), and * is bilinear, that is, a linear map from V ⊗ V into V. A second bilinear map is deduced from the first by permuting the entries:

      [1.58]

      It also makes sense to look at tri-, quadri- and multi-linear operations, that is, linear maps from V^⊗n to V for any V. For example, it is very easy to produce 12 tri-linear maps starting with the bilinear map * by considering: