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

By Loday J.-L., Vallette B.

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Its product (concatenation) is denoted by µ. Let us denote by ∆ the deconcatenation coproduct. Show that they satisfy the following compatibility relation: ccc   cc = −   c cc cc  c  c + . ccc  + ccc    ¯ be the reduced diagonal. What Let T (V ) be the augmentation ideal and let ∆ ¯ is the compatibility relation between µ and ∆ on T (V ) ? 8. 13. Baker-Campbell-Hausdorff. Show that the polynomials Hn (x, y) showing up in the BCH formula exp(x) exp(y) = exp(x + y + · · · + Hn (x, y) + · · · ) (1) can be computed out of the Eulerian idempotent en (see [Lod94]).

Let A be a dga algebra and let N be a chain complex. There is a one-to-one correspondence between A-derivations on the free A-module N ⊗ A and linear maps from N to N ⊗ A: Der(N ⊗ A) ∼ = Hom(N, N ⊗ A), df = (Id ⊗ µ) ◦ (f ⊗ Id) ↔ f . Dually, let C be a dga coalgebra and let N be a chain complex. There is a one-toone correspondence between coderivations on the cofree C-module C ⊗ N and linear maps from C ⊗ N to N : Coder(C ⊗ N ) ∼ = Hom(C ⊗ N, N ), df = (Id ⊗ f ) ◦ (∆ ⊗ Id) ↔ f . Proof. First statement.

When n ranges over Z we get a pre-Lie algebra whose associated Lie algebra is well-known. For K = R it is the Lie algebra of polynomial vector fields over the circle (Virasoro algebra without center). For K being a finite field it is called the Witt algebra. (c) Derivations. Let Di , i = 1, . . , k, be commuting derivations of a commutative algebra A. On the free A-module spanned by the Di ’s one defines {aDi , bDj } := bDj (a)Di . Since we assumed that the derivations are commuting, it is immediate to verify that this is a pre-Lie product.

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