On nonsingular two-step nilpotent Lie algebras

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.