By Jean-Louis Loday

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KAPRANOV, Koszul duality for operads, Duke Math. J. 76 (1994), 203–272. [Go] F. GOICHOT, Un th´eor`eme de Milnor-Moore pour les alg`ebres de Leibniz, [Ha] P. HANLON, The fixed point partition lattices, Pacific J. Math. 2, (1981), 319–341. [Hi] P. HIGGINS, Thesis, Oxford (UK). N. INASSARIDZE, Homotopy of pseudosimplicial groups, nonabelian derived functors, and algebraic K-theory. (Russian) Mat. Sb. ) 98(140) (1975), no. 3(11), 339–362, 495. [KS] Y. KOSMANN-SCHWARZBACH, From Poisson algebras to Gerstenhaber algebras.

N). In particular, θn1 (1, . . , n) = (1, . . , n) and θnn (1, . . , n) = (n, . . , 1). The map Θn : K[Cn ] ⊗ R⊗n → R⊗n defined by Θn ([r] ⊗ (x1 , · · · xn )) := θnr (x1 , · · · xn ) is the chain complex map Θ∗ : C∗Dend (RDend ) −→ C∗Zinb (R) induced by the operad morphism Dend → Zinb. Proof. This is a consequence of the explicit description of the morphism of Leibniz algebras Leib(V ) → Dias(V )Leib , cf. Appendix B5e. Observe that θnr is the sum of the signed action of ( nr ) permutations.

The map ψ can be considered as the restriction to the vertices of a cellular map between polytopes. , n} let us define a poset as follows. An element of the poset is an ordered partition of < n >. The element X is less than the element Y if Y can be obtained from X by reuniting successive subsets. For instance {(35)(2)(14)(7)(6)} < {(235)(14)(67)}. Note that the minimal elements are precisely the permutations. n)} from the poset, then the geometric realization is homeomorphic to the sphere S n−2 (it is called the permutohedron).