Generic properties of 2-step nilpotent Lie algebras and torsion-free groups

TL;DR

This paper investigates the typical structural properties of 2-step nilpotent Lie algebras and torsion-free nilpotent groups, identifying generic features and conditions for certain algebraic behaviors using algebraic geometry methods.

Contribution

It introduces a framework for defining generic properties of 2-step nilpotent Lie algebras and computes key structural bounds, also extending results to torsion-free nilpotent groups of class 2.

Findings

Maximal dimension of abelian subalgebras in generic Lie algebras

Conditions for generic Lie algebras to lack surjective homomorphisms onto certain non-abelian Lie algebras

Analogous properties established for torsion-free nilpotent groups of class 2

Abstract

To define the notion of a generic property of finite dimensional 2-step nilpotent Lie algebras we use standard correspondence between such Lie algebras and points of an appropriate algebraic variety, where a negligible set is one contained in a proper Zariski-closed subset. We compute the maximal dimension of an abelian subalgebra of a generic Lie algebra and give a sufficient condition for a generic Lie algebra to admit no surjective homomorphism onto a non-abelian Lie algebra of a given dimension. Also we consider analogous questions for finitely generated torsion free nilpotent groups of class 2.