Cocommutative Com-PreLie bialgebras

TL;DR

This paper classifies connected, cocommutative Com-PreLie bialgebras over fields of characteristic zero, identifying main families based on symmetric and group algebra structures, and explores their extensions over group and tensor product algebras.

Contribution

It provides a complete classification of connected, cocommutative Com-PreLie bialgebras, including new families and special cases involving group and tensor product algebras.

Findings

Main family: symmetric algebras on any-dimensional space V.

Additional family exists only if V is one-dimensional.

Extensions over group algebras and tensor products are characterized.

Abstract

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.