  A Com-PreLie bialgebra is a commutative bialgebra with an extra preLie
product satisfying some compatibilities with the product and coproduct. We here
give a classification of connected, cocommutative Com-PreLie bialgebras over a
field of characteristic zero: we obtain a main family of symmetric algebras on
a space V of any dimension, and another family available only if V is
one-dimensional. We also explore the case of Com-PreLie bialgebras over a group
algebra and over a tensor product of a group algebra and of a symmetric
algebra.
