A classification of pre-Lie $H$-pseudoalgebras of low ranks

TL;DR

This paper classifies low-rank pre-Lie pseudoalgebras over the universal enveloping algebra of a finite-dimensional Lie algebra, focusing on rank 1 and rank 2 cases, including their associativity properties.

Contribution

It provides a detailed classification of low-rank pre-Lie $H$-pseudoalgebras over $U( ext{Lie algebra})$, including explicit structures and associativity conditions.

Findings

Classified rank 1 pre-Lie $H$-pseudoalgebras.

Introduced and classified a class generated by two rank 1 pseudoalgebras.

Presented explicit associativity classification for the constructed pseudoalgebras.

Abstract

Let $H=U(\delta)$ be the universal enveloping algebra of finite dimension Lie algebra $\delta$. The central result of the paper is the classification of pre-Lie $H$-pseudoalgebras of low ranks over the Hopf algebra $H$. We firstly study pre-Lie pseudoalgebras that are free of rank $1$ over $H$. Then we introduce and classify a class of pre-Lie $H$-pseudoalgebras $\mathcal{P}$ which are generated by two pre-Lie pseudoalgebras of rank $1$. Finally, the associativity of $\mathcal{P}$ is also considered and a explicit assification is presented.