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By Loday J.-L., Vallette B.

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Example text

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.

TWISTING MORPHISMS One can extend this functor to the case where (A, dA ) is an augmented differential graded algebra. Indeed, the differential dA : A → A induces a differential on A⊗n by n d1 := (id, . . , id, dA , id, . . , id). i=1 ¯ Since µA is a morphism of differential We denote by d1 the differential on T c (sA). graded vector spaces, one can check that d1 and d2 anticommute: d1 ◦d2 +d2 ◦d1 = 0. The chain complex associated to the total differential d1 + d2 is called the bar construction of the augmented differential graded algebra ¯ dBA = d1 + d2 BA := T c (sA), The analogous construction in algebraic topology (classifying space of a topological group) is also called bar construction and denoted B.

Then τ is extended by linearity. The category gVect, equipped with this symmetry, is called the category of sign-graded vector spaces. Observe that there is another choice of symmetry: τ (v ⊗w) := w ⊗v. Equipped with this symmetry gVect is simply called the category of graded vector spaces. 3. Koszul sign rule. When working in the symmetric monoidal category of sign-graded vector spaces with switching map τ , there are signs involved in the formulas. For instance, if A is a graded algebra, then the product in A ⊗ A is given by (x ⊗ y)(x ⊗ y ) = (−1)|y||x | (xx ) ⊗ (yy ).

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