We give a new presentation of the Lie cooperad as a quotient of the graph cooperad, a presentation which is not linearly dual to any of the standard presentations of the Lie operad. We use this presentation to explicitly compute duality between Lie algebras and coalgebras, to give a new presentation of Harrison homology, and to show that Lyndon words yield a canonical basis for cofree Lie coalgebras.
Homology, Homotopy and Applications, Vol. 13 (2011), No. 2, pp.263-292.