Degenerations of pre-Lie algebras

TL;DR

This paper studies how pre-Lie algebra structures on a fixed vector space can degenerate, analyzing orbit closures under group actions, and classifies all such degenerations in 2-dimensional complex cases.

Contribution

It introduces the concept of pre-Lie algebra degenerations, establishes fundamental invariants and criteria, and classifies all orbit closures for 2-dimensional complex pre-Lie algebras.

Findings

Defined pre-Lie algebra degenerations and orbit closures.

Established invariants and criteria for degenerations.

Classified all orbit closures in 2-dimensional complex case.

Abstract

We consider the variety of pre-Lie algebra structures on a given n-dimensional vector space. The group GL_n(K) acts on it, and we study the closure of the orbits with respect to the Zariski topology. This leads to the definition of pre-Lie algebra degenerations. We give fundamental results on such degenerations, including invariants and necessary degeneration criteria. We demonstrate the close relationship to Lie algebra degenerations. Finally we classify all orbit closures in the variety of complex 2-dimensional pre-Lie algebras.