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On extensions of Frobenius-Kähler and Sasakian Lie algebras

M. C. Rodríguez-Vallarte, G. Salgado, O. A. Sánchez-Valenzuela

TL;DR

This work investigates how Sasakian and Frobenius-Kähler Lie algebras behave under double extensions and derivation extensions. It derives explicit necessary-and-sufficient conditions for when a double extension of a Sasakian Lie algebra remains Sasakian, expressed in terms of cocycles, derivations, and lifted Sasakian data. It further establishes constructive links between Frobenius-Kähler and Sasakian structures: FK to Sasakian extensions via derivations commuting with the complex structure, and Sasakian to FK extensions via appropriately chosen derivations and lifted endomorphisms. The paper includes concrete, low-dimensional examples that illustrate the constructions and clarify when reversed double extensions are possible or obstructed, highlighting the interplay between central extensions, derivations, and geometric structures on Lie algebras.

Abstract

Extensions of Lie algebras equipped with Sasakian or Frobenius-Kähler geometrical structures are studied. Conditions are given so that a double extension of a Sasakian Lie algebra be Sasakian again. Conditions are also given for obtaining either a Sasakian or a Frobernius-Kähler Lie algebra upon respectively extending a Frobernius-Kähler or a Sasakian Lie algebra by adjoining a derivation of the source algebra. Low-dimensional examples are included.

On extensions of Frobenius-Kähler and Sasakian Lie algebras

TL;DR

This work investigates how Sasakian and Frobenius-Kähler Lie algebras behave under double extensions and derivation extensions. It derives explicit necessary-and-sufficient conditions for when a double extension of a Sasakian Lie algebra remains Sasakian, expressed in terms of cocycles, derivations, and lifted Sasakian data. It further establishes constructive links between Frobenius-Kähler and Sasakian structures: FK to Sasakian extensions via derivations commuting with the complex structure, and Sasakian to FK extensions via appropriately chosen derivations and lifted endomorphisms. The paper includes concrete, low-dimensional examples that illustrate the constructions and clarify when reversed double extensions are possible or obstructed, highlighting the interplay between central extensions, derivations, and geometric structures on Lie algebras.

Abstract

Extensions of Lie algebras equipped with Sasakian or Frobenius-Kähler geometrical structures are studied. Conditions are given so that a double extension of a Sasakian Lie algebra be Sasakian again. Conditions are also given for obtaining either a Sasakian or a Frobernius-Kähler Lie algebra upon respectively extending a Frobernius-Kähler or a Sasakian Lie algebra by adjoining a derivation of the source algebra. Low-dimensional examples are included.
Paper Structure (10 sections, 18 theorems, 40 equations)

This paper contains 10 sections, 18 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Frobenius-Kähler and Sasakian Lie Algebras
  3. Basic definitions
  4. Double extensions with integrable $J$'s
  5. Double extensions of Kähler and Sasakian Lie algebras
  6. Sasakian double extension
  7. Extensions of Frobenius-Kähler and Sasakian Lie algebras
  8. From Frobenius-Kähler to Sasakian Lie algebras
  9. From Sasakian to Frobenius-Kähler Lie algebras
  10. Examples

Key Result

Theorem 2.2

Let $(\xi,\alpha,\Phi)$ be a Sasa-kian structure on a Lie algebra $(\mathfrak{g},[\,\cdot\,,\,\cdot\,])$ with non-trivial one-dimensional center generated by $\xi$, and let $\mathfrak{h}=\operatorname{Ker}(\alpha)$. Let $J =\Phi\vert_{\mathfrak{h}}$ and let $\omega$ be obtained through $d\alpha$ fro

Theorems & Definitions (35)

  • Remark 2.1
  • Theorem 2.2: See A-F-V
  • Theorem 2.3: See Smo
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • ...and 25 more