Cocommutative Com-PreLie bialgebras
Loïc Foissy
TL;DR
This work classifies connected cocommutative Com-PreLie bialgebras over a field of characteristic zero, showing that they are either symmetric algebras $S(V,f,\lambda)$ with an explicit $\bullet$-product or, in the one-dimensional primitive case, the special family $\mathfrak{g}^{(1)}(1,a,1)$. A constructive approach via two operators $\partial$ and $\phi$ yields the $S(V,f,\lambda)$ family with $1\bullet x=0$ and $x\bullet x_1\ldots x_k=\sum_{I\subsetneq [k]} |I|! \lambda^{|I|} f(x)\prod_{i\in I}f(x_i)\prod_{i\notin I}x_i$, and verifies Leibniz, pre-Lie, and coproduct compatibility. The paper further classifies homogeneous pre-Lie products on $\mathbb{K}[X]$, identifying four families, and analyzes their underlying Lie algebras, including Faà di Bruno structures and degree-derivation components. Extending to non-connected settings, the authors determine all pre-Lie structures on group Hopf algebras $\mathbb{K}G$ and on $\mathbb{K}G\cdot S(V)$, with complete parameterizations for $G=\mathbb{Z}$ and a general set of compatibility conditions for abelian groups, culminating in a comprehensive classification (Theorem 23) for $\mathbb{K}G\cdot S(V)$.
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.