Degenerations of Lie algebras and geometry of Lie groups

By Jorge Lauret. 2003

84 citations recorded by the source corpus.

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 2 S L ( m )⊗S L ( n ).

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ba7e4a252fc8f9c582bdfa38eff51ad32b412dbe

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In Applied Mathematics

1. Y. Volkov
2. A. Takahara
3. Hiroshi Kodama
4. Ivan Kaygorodov
5. H. Tamaru

In Astronomy and Astrophysics

1. Y. Volkov
2. A. Takahara
3. Hiroshi Kodama
4. Ivan Kaygorodov
5. H. Tamaru

In Geometry and Topology

1. Y. Volkov
2. Ivan Kaygorodov
3. A. Takahara
4. Hiroshi Kodama
5. H. Tamaru

Citations in the corpus

• The space of left-invariant metrics on a Lie group up to isometry and scaling (2011, background): “Note that the proof by Lauret in [ 9 ] uses degenerations of bracket products and curvature pinching.”
• The Variety of Two-dimensional Algebras Over an Algebraically Closed Field (2017, background): “Degenerations are an interesting subject, which was studied in various papers (see, for example, [3–7, 10–12, 14–16, 18, 20, 21, 25–27, 29]).”
• Degenerations of nilpotent algebras (2019, background): “INTRODUCTION There are many results related to the algebraic and geometric classification of low-dimensional algebras in the varieties of Jordan, Lie, Leibniz and Zinbiel algebras; for algebraic classifications see, for example, [1,7,13,16–20,24,26,33,34,36,37,39,42]; for geometric classifications a”
• DEGENERATIONS OF 7-DIMENSIONAL NILPOTENT LIE ALGEBRAS (2004, unknown): “For some Lie algebras, such as 3 ⊕ km, where 3 is the 3-dimensional Heisenberg Lie algebra, this is the only possible degeneration, see Lauret (2003).”