Degenerations of Lie algebras and geometry of Lie groups

Abstract

Each point of the variety of real Lie algebras is naturally identified with a left invariant Riemannian metric on a Lie group. We study the interplay between invariant-theoretic and Riemannian aspects of this variety. In particular, using the special critical point behavior of certain natural functional on the variety, we determine all the Lie groups which can be endowed with only one left invariant metric up to isometry and scaling, proving first that they correspond to Lie algebras whose only degeneration is to the abelian one. We also find all the Lie algebras which degenerate to the Lie algebra of the hyperbolic space, and all the possible degenerations for 3-dimensional real Lie algebras, by using well known descriptions of left invariant metrics satisfying some pinching curvature conditions. Finally, as another interaction, the closed S L( n)-orbits on the variety are classified, and explicit curves of Einstein solvmanifolds are provided by using curves of closed orbits of the representation Λ 2S L( m)⊗S L( n).