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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.
Loïc Foissy (Fri,) studied this question.