Cocommutative Com-PreLie bialgebras

A Com-PreLie bialgebra is a commutative bialgebra with an extra pre-Lie product satisfying some compatibilities with the product and the 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 [Formula: see text] of any dimension, and another family available only if [Formula: see text] 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.