On five dimensional Sasakian Lie algebras with trivial center

TL;DR

This paper classifies five-dimensional Sasakian Lie algebras with trivial center, proving they are all $ ext{phi}$-symmetric, and constructs a family of contact metric structures with specific invariants on certain Lie groups.

Contribution

It characterizes five-dimensional Sasakian Lie algebras with trivial center as $ ext{phi}$-symmetric and constructs new contact metric structures with all Boeckx invariants less than -1.

Findings

All such Lie algebras are $ ext{phi}$-symmetric.

Constructed a family of contact metric $(k, extmu)$ structures.

Boeckx invariants cover all values less than -1.

Abstract

We show that every five-dimensional Sasakian Lie algebra with trivial center is $\varphi$-symmetric. Moreover starting from a particular Sasakian structure on the Lie group $SL(2,\mathbb{R})\times\text{Aff}(\mathbb{R})$ we obtain a family of contact metric $(k,\mu)$ structures whose Boeckx invariants assume all values less than $-1$.