The necklace Lie coalgebra and renormalization algebras

We give a natural monomorphism from the necklace Lie coalgebra, defined forany quiver, to Connes and Kreimer's Lie coalgebra of trees, and extend this toa map from a certain quiver-theoretic Hopf algebra to Connes and Kreimer'srenormalization Hopf algebra, as well as to pre-Lie versions. These results aredirect analogues of Turaev's results in 2004, by replacing algebras of loops onsurfaces with algebras of paths on quivers. We also factor the morphism throughan algebra of chord diagrams and explain the geometric version. We then explainhow all of the Hopf algebras are uniquely determined by the pre-Lie structures,and discuss noncommutative versions of the Hopf algebras.