On nonsingular two-step nilpotent Lie algebras

TL;DR

This paper investigates nonsingular 2-step nilpotent Lie algebras, focusing on their classification, canonical inner products, automorphism groups, and using tools like the moment map and Pfaffian form for analysis.

Contribution

It introduces new invariants and methods for classifying nonsingular 2-step nilpotent Lie algebras, including the use of the moment map and Pfaffian form.

Findings

Classification invariants for nonsingular algebras

Existence of canonical inner products (nilsolitons)

Automorphism groups with maximality properties

Abstract

A 2-step nilpotent Lie algebra n is called nonsingular if ad(X): n --> [n,n] is onto for any X not in [n,n]. We explore nonsingular algebras in several directions, including the classification problem (isomorphism invariants), the existence of canonical inner products (nilsolitons) and their automorphism groups (maximality properties). Our main tools are the moment map for certain real reductive representations, and the Pfaffian form of a 2-step algebra, which is a positive homogeneous polynomial in the nonsingular case.