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A homotopy for a complex of free Lie algebras

Using the Guichardet construction, we compute the cohomology groups of a complex of free Lie algebras introduced by Alekseev and Torossian. | Turk J Math 36 (2012) , 59 – 65. ¨ ITAK ˙ c TUB doi:10.3906/mat-1101-76 A homotopy for a complex of free Lie algebras Mich`ele Vergne Abstract Using the Guichardet construction, we compute the cohomology groups of a complex of free Lie algebras introduced by Alekseev and Torossian. 1. Introduction In their study of the relation between the KV-conjecture and Drinfeld’s associators, Alekseev and Torossian [1] studied the Eilenberg-MacLane differential δA : Ln → Ln+1 , where Ln is the free Lie algebra in n variables, and computed the cohomology groups of δA in dimensions 1, 2 . Following the construction of Guichardet [2] (see also [3]), we remark that the complex δA is acyclic, except in dimensions 1, 2 , where the cohomology is of dimension 1 . We also identify the cohomology groups of a similar complex δA : Tn → Tn+1 , where Tn is the free associative algebra in n variables: the cohomology is of dimension 1 in any degree. The Guichardet construction provides an explicit homotopy. Alekseev and Torossian used the computations in dimension 2 to deduce the existence of a solution to the KV problem from the existence of an associator. A simple by-product of their computation is the existence and the uniqueness of the Campbell-Hausdorff formula. We do not have any other application of the computations of higher cohomologies. In this note, we start with a review of the construction of Guichardet. Then we adapt it to free associative algebras and free Lie algebras. I am thankful to the referee for his careful reading. 2. The Guichardet construction Let V be a finite dimensional real vector space. Let F n be the space of polynomial functions f on V ⊕ V ⊕ · · · ⊕ V . An element f of F n is written as f(v1 , v2 , . . . , vn ). Define (δn f)(v1 , . . . , vn+1 ) = n (−1)i f(v1 , v2 , . . . , vi−1 , vˆi , vi+1 , . . . , vn ). i=1 For example: (δ1 f)(v1 , v2 ) = −f(v2 ) + f(v1 ) 59 VERGNE (δ2 f)(v1 , v2 , v3 ) = −f(v2 , v3 ) + f(v1 , v3 ) − f(v1 , v2 ). We define F