Outer restricted derivations of nilpotent restricted Lie algebras

TL;DR

This paper proves that most finite-dimensional nilpotent restricted Lie algebras over a field of prime characteristic have an outer restricted derivation with square zero, except for specific small or special cases.

Contribution

It establishes a restricted analogue of a classical Lie algebra result, identifying conditions under which outer restricted derivations with square zero exist.

Findings

Most finite-dimensional nilpotent restricted Lie algebras have such derivations.

Exceptions include tori, one-dimensional, or specific low-dimensional cases.

The result parallels classical Lie and group-theoretic theorems.

Abstract

In this paper we prove that every finite-dimensional nilpotent restricted Lie algebra over a field of prime characteristic has an outer restricted derivation whose square is zero unless the restricted Lie algebra is a torus or it is one-dimensional or it is isomorphic to the three-dimensional Heisenberg algebra in characteristic two as an ordinary Lie algebra. This result is the restricted analogue of a result of T\^og\^o on the existence of nilpotent outer derivations of ordinary nilpotent Lie algebras in arbitrary characteristic and the Lie-theoretic analogue of a classical group-theoretic result of Gasch\"utz on the existence of $p$-power automorphisms of $p$-groups. As a consequence we obtain that every finite-dimensional non-toral nilpotent restricted Lie algebra has an outer restricted derivation.