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A complete classification of symplectic forms on six-dimensional Frobeniusian real Lie algebras

T. Aït Aissa, S. Elbourkadi, M. W. Mansouri

Abstract

In this paper, we give a complete classification of symplectic structures on six-dimensional Frobeniusian solvable Lie algebras, up to symplectomorphism. We provide a scheme to classify the isomorphism classes of six-dimensional Frobeniusian solvable Lie algebras whose exact form has a Lagrangian ideal. We complete our classification by considering Frobeniusian solvable Lie algebras without Lagrangian ideal.

A complete classification of symplectic forms on six-dimensional Frobeniusian real Lie algebras

Abstract

In this paper, we give a complete classification of symplectic structures on six-dimensional Frobeniusian solvable Lie algebras, up to symplectomorphism. We provide a scheme to classify the isomorphism classes of six-dimensional Frobeniusian solvable Lie algebras whose exact form has a Lagrangian ideal. We complete our classification by considering Frobeniusian solvable Lie algebras without Lagrangian ideal.
Paper Structure (17 sections, 26 theorems, 95 equations, 14 tables)

This paper contains 17 sections, 26 theorems, 95 equations, 14 tables.

Table of Contents

  1. Introduction and main result
  2. Lagrangian extensions of flat Lie algebras
  3. Change of flat Lie algebra
  4. Exact Lagrangian extension
  5. Bijective $1$-Cocycles
  6. Three-dimensional flat Lie algebras
  7. Six-dimensional Frobeniusian Lie algebras
  8. Classification of Frobeniusian Lie algebras with Lagrangian ideals
  9. Appendix A
  10. Cohomological properties of flat Lie algebras
  11. Symplectic structures on six-dimensional Frobeniusian Lie algebras with Lagrangian ideal
  12. Symplectic structures on six-dimensional Frobeniusian Lie algebras without Lagrangian ideal
  13. The flat torsion-free connections that are derived from the symplectic structures
  14. Appendix B
  15. Isomorphisms of ${\mathfrak{g}}_{6,j}$ to decomposable Lie algebras
  16. ...and 2 more sections

Key Result

Theorem 1

Let $({\mathfrak{g}},\omega)$ be a six-dimensional Frobeniusian Lie algebra. Then, $({\mathfrak{g}},\omega)$ is isomorphic to exactly one of the following symplectic Lie algebras$:$

Theorems & Definitions (53)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Lemma 2
  • Example 1
  • Proposition 3
  • ...and 43 more