Mathematical Research Letters

Volume 21 (2014)

Number 3

On non-singular 2-step nilpotent Lie algebras

Pages: 553 – 583

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n3.a11

Authors

Jorge Lauret (FaMAF and CIEM, Universidad Nacional de Córdoba, Ciudad Universitaria, Córdoba, Argentina)

David Oscari (FaMAF and CIEM, Universidad Nacional de Córdoba, Ciudad Universitaria, Córdoba, Argentina)

Abstract

A 2-step nilpotent Lie algebra $\mathfrak{n}$ is called non-singular if ad $X : \mathfrak{n} \to [ \mathfrak{n}, \mathfrak{n} ]$ is onto for any $X \notin [ \mathfrak{n}, \mathfrak{n} ]$. We explore non-singular 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 non-singular case.

Full Text (PDF format)