Left invariant flat projective structures on Lie groups and prehomogeneous vector spaces

2014/06/13 by Hironao Kato, Kato, Hironao
Mathematics · #11S90 #53A20 #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:11S90 #msc:53A20

paper · pdf · doi:10.48550/arxiv.1406.3426

33 pages

arxiv created 2014/06/13 · openalex publication_date 2014/06/13 · arxiv updated 2014/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show the correspondence between left invariant flat projective structures on Lie groups and certain prehomogeneous vector spaces. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex Lie groups admitting irreducible left invariant flat complex projective structures. As a result, direct sums of special linear Lie algebras sl(2) ⊕ sl(m₁) ⊕ ⋯ ⊕ sl(m_k) admit left invariant flat complex projective structures if the equality 4 + m₁² + ⋯ + m_k² -k - 4 m₁ m₂ ⋯ m_k = 0 holds. These contain sl(2), sl(2) ⊕ sl(3), sl(2) ⊕ sl(3) ⊕ sl(11) for example.

Related

• Low dimensional Lie groups admitting left invariant flat projective or affine structures
• Flat Lie groups, Frobenius Lie algebras and étale prehomogeneous vector spaces for reductive Lie groups
• Flat Affine or Projective Geometries on Lie Groups
• Automorphisms of projective structures
• Lie Algebras : Classifications, Deformations, Rigidity and Differential Geometry