On extensions of Frobenius-K\"ahler and Sasakian Lie algebras

TL;DR

This paper investigates how certain geometric structures on Lie algebras, like Sasakian and Frobenius-Kähler, can be extended while preserving their properties, providing conditions and examples for such extensions.

Contribution

It establishes conditions under which extensions of Sasakian and Frobenius-Kähler Lie algebras retain their structures, including new criteria and low-dimensional examples.

Findings

Double extensions of Sasakian Lie algebras can remain Sasakian under specific conditions.

Extensions by derivations can produce new Sasakian or Frobenius-Kähler Lie algebras.

Explicit low-dimensional examples illustrate the theoretical results.

Abstract

Extensions of Lie algebras equipped with Sasakian or Frobenius-K\"ahler 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-K\"ahler Lie algebra upon respectively extending a Frobernius-K\"ahler or a Sasakian Lie algebra by adjoining a derivation of the source algebra. Low-dimensional examples are included.