Degenerations of pre-Lie algebras

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 two-dimensional pre-Lie algebras. (C) 2009 American Institute of Physics. [doi:10.1063/1.3246608]