Automorphisms of non-singular nilpotent Lie algebras

Abstract

For a real, non-singular, 2-step nilpotent Lie algebra $\mathfrak{n}$, the group \Aut(\mathfrak{n})/\Aut_0(\mathfrak{n})$,where$\Aut_0(\mathfrak{n})$isthegroupofautomorphismswhichacttriviallyonthecenter,isthedirectproductofacompactgroupwiththe1-dimensionalgroupofdilations.Maximalityofsomeautomorphismsgroupsof$\mathfrak{n}$followsandisrelatedtohowcloseis$\mathfrak{n}$tobeingofHeisenbergtype.Forexample,atleastwhenthedimensionofthecenteristwo,$\dim \Aut(\mathfrak{n})$ismaximalifandonlyif$\mathfrak{n}$istype$H$. The connection with fat distributions is discussed.