A nonisomorphism theorem for cofree Lie coalgebras (Q1102370)

Let V be a vector space over a field k. Let \(L\subset V\) be the cofree Lie coalgebra on V. \(L\subset V\) is a Lie coalgebra in the sense that it is a coalgebra satisfying rules dual to the anticommutativity and Jacobi identity satisfied by a Lie algebra. It is cofree in the sense that there is a linear map from \(L\subset V\) to V satisfying a universal property dual to the one satisfied by the inclusion of a Lie algebra into its universal enveloping algebra. A Lie coalgebra is called covered if it is the homomorphic image of a Lie coalgebra \(C^-\) arising from a coassociative coalgebra C by twisting its comultiplication. There is also a cofree covered Lie coalgebra on V, and a canonical injection of it into \(L\subset V\). The author proves that this injection is surjective if and only if V is at most one-dimensional over k.