Moduli of Lie p-algebras

TL;DR

This paper investigates the structure and properties of moduli spaces of finite-dimensional Lie p-algebras, revealing new cases of p-mappings, and analyzing their smoothness and categorical equivalences, especially for rank 3 cases.

Contribution

It introduces new results on the affine fibration of the forgetful map, identifies a new case of p-mapping existence, and constructs the moduli space for rank 3 Lie p-algebras over Z.

Findings

The forgetful map from Lie p-algebras to Lie algebras is an affine fibration.

A new case of p-mapping existence is identified.

The moduli space of rank 3 Lie p-algebras is constructed and analyzed for smoothness.

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.