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The moduli spaces of presymplectic forms on almost abelian Lie algebras

Luis Pedro Castellanos Moscoso

TL;DR

This work analyzes left-invariant presymplectic forms on almost abelian Lie groups by translating the geometric problem into a real and complex matrix-congruence problem tied to the Jordan normal form of the adjoint action. It establishes precise existence criteria for presymplectic forms of a given rank $R$, expressed via structured skew blocks satisfying $B\mathcal J_N+(\mathcal J_N)^tB=0$, and decomposes the moduli space into tractable real and complex parts with canonical representatives. A key result is that the moduli space of symplectic forms on any almost abelian Lie algebra is finite, with all symplectic forms equivalent to a permutation of a canonical $2$-form, determined by the real and complex eigenvalue data. The paper also develops a rich matrix-theoretic framework, including $N$-upper Toeplitz and lower Hankel structures, to classify these forms and their equivalence classes, thereby linking presymplectic geometry to canonical matrix-congruence problems of independent interest.

Abstract

We obtain necessary and sufficient conditions to determine the existence of presymplectic forms of a given rank on all almost abelian Lie algebras. We also study the moduli space of presymplectic forms (this is the set of all closed 2-forms of a given rank under a certain natural equivalence relation) on almost abelian Lie algebras. Most importantly we show that for any almost abelian Lie algebra its moduli space of symplectic forms is finite. Moreover we show that up to such natural equivalence all symplectic forms are permutations of a canonical 2-form. The important step in the proof is obtaining canonical representatives for a certain congruence of matrices, which is of some interest for matrix theory on its own.

The moduli spaces of presymplectic forms on almost abelian Lie algebras

TL;DR

This work analyzes left-invariant presymplectic forms on almost abelian Lie groups by translating the geometric problem into a real and complex matrix-congruence problem tied to the Jordan normal form of the adjoint action. It establishes precise existence criteria for presymplectic forms of a given rank , expressed via structured skew blocks satisfying , and decomposes the moduli space into tractable real and complex parts with canonical representatives. A key result is that the moduli space of symplectic forms on any almost abelian Lie algebra is finite, with all symplectic forms equivalent to a permutation of a canonical -form, determined by the real and complex eigenvalue data. The paper also develops a rich matrix-theoretic framework, including -upper Toeplitz and lower Hankel structures, to classify these forms and their equivalence classes, thereby linking presymplectic geometry to canonical matrix-congruence problems of independent interest.

Abstract

We obtain necessary and sufficient conditions to determine the existence of presymplectic forms of a given rank on all almost abelian Lie algebras. We also study the moduli space of presymplectic forms (this is the set of all closed 2-forms of a given rank under a certain natural equivalence relation) on almost abelian Lie algebras. Most importantly we show that for any almost abelian Lie algebra its moduli space of symplectic forms is finite. Moreover we show that up to such natural equivalence all symplectic forms are permutations of a canonical 2-form. The important step in the proof is obtaining canonical representatives for a certain congruence of matrices, which is of some interest for matrix theory on its own.
Paper Structure (20 sections, 49 theorems, 272 equations)

This paper contains 20 sections, 49 theorems, 272 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Setting
  3. Left-invariant presymplectic $2$-forms
  4. Almost abelian Lie algebras
  5. The sets $\Omega^2_{R,\,closed}$ and $\mathfrak{P}\Omega^2_{R,\,closed}$ for almost abelian Lie algebras.
  6. The moduli space for Almost abelian Lie algebras.
  7. A Matrix problem
  8. The sets $\mathcal{T}_{\mathcal{J}_{N_{\mathbb{R}}}}$ and $\mathcal{H}_{\mathcal{J}_{{N}_{\mathbb{R}}}}^{K}$: real eigenvalues.
  9. Some special matrices and its properties
  10. The solutions of equations (\ref{['permutationequation']}) and (\ref{['transposepermutationequation']})
  11. The quotient space
  12. Some more matrices and notation.
  13. The set $\dot{\mathcal{H}}_{\mathcal{J}_{N_{\mathbb{R}}}}^{K}/_{\text{N-blockstar cong}}\dot{\mathcal{T}}_{\mathcal{J}_{N_{\mathbb{R}}}}$
  14. The sets $\mathcal{T}_{\mathcal{J}_{N_{\mathbb{C}}}}$ and $\mathcal{ H}_{\mathcal{J}_{N_{\mathbb{C}}}}^{W}$: Complex eigenvalues
  15. Some special matrices and its properties
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\mathfrak{g}$ be an almost abelian Lie algebra. Then there exist a presymplectic $2$ form of rank $R$ if "enough" eigenvalues of the associated matrix to $\mathfrak{g}$ come in positive-negative pairs.

Theorems & Definitions (102)

  • Theorem : Main result 1
  • Theorem : Main result 2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: cf. SymplecticLieGroups, Chapter 0
  • Remark 2.4
  • Definition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Proposition 3.4
  • ...and 92 more