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

TL;DR

The paper addresses the classification of low-rank pre-Lie $H$-pseudoalgebras over the cocommutative Hopf algebra $H=U(\delta)$. It begins with a rank-one analysis, showing that the pseudo-product on a free rank-one module is determined by $\alpha\in H\otimes H$ and that, for $H=U(\delta)$, the solutions satisfy $\alpha=1\otimes s+t\otimes1$ with $s\in\delta$ and $t\in\mathbf{k}$, yielding left and right pre-Lie structures when appropriate. It then treats rank-two algebras $\mathcal{P}=He_1\oplus He_2$ generated by two rank-one pieces, providing a comprehensive classification into multiple isomorphism types depending on parameters $(s_1,t_1,s_2,t_2)$, and derives associativity criteria culminating in a detailed list of associative forms. The results give a systematic framework for constructing and recognizing pre-Lie pseudoalgebras in the pseudotensor setting, with implications for deformation theory, representation theory of Lie pseudoalgebras, and potential applications in mathematical physics.

Abstract

Let $H=U(δ)$ be the universal enveloping algebra of finite dimension Lie algebra $δ$. 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.