Moduli of Lie p-algebras

Abstract

In this paper, we study moduli spaces of finite-dimensional Lie algebras with flat center, proving that the forgetful map from Lie p-algebras to Lie algebras is an affine fibration, and we point out a new case of existence of a p-mapping. Then we illustrate these results for the special case of Lie algebras of rank 3, whose moduli space we build and study over Z. We extend the classical equivalence of categories between locally free Lie p-algebras of finite rank with finite locally free group schemes of height 1, showing that the centers of these objects correspond to each other. We finish by analysing the smoothness of the moduli of p-Lie algebras of rank 3, in particular identifying some smooth components.