The Grunewald-O'Halloran conjecture for nilpotent Lie algebras of rank >= 1

TL;DR

This paper proves the Grunewald-O'Halloran conjecture for a class of nilpotent Lie algebras admitting semisimple derivations, showing they are degenerations of non-isomorphic algebras, and provides explicit degenerations in dimension 7.

Contribution

It establishes the conjecture for nilpotent Lie algebras with semisimple derivations and constructs explicit degenerations in dimension 7.

Findings

Proved the conjecture for algebras with semisimple derivations.

Constructed explicit degenerations for all 7-dimensional nilpotent Lie algebras.

Remaining open for characteristically nilpotent Lie algebras.

Abstract

Grunewald and O'Halloran conjectured in 1993 that every complex nilpotent Lie algebra is the degeneration of another, non isomorphic, Lie algebra. We prove the conjecture for the class of nilpotent Lie algebras admitting a semisimple derivation, remaining open for the class of characteristically nilpotent Lie algebras. In dimension 7, where the first characteristically nilpotent Lie algebras appear, we prove the conjecture and we also exhibit explicit nontrivial degenerations to every 7-dimensional nilpotent Lie algebra.