On five dimensional Sasakian Lie algebras with trivial center

Abstract

We show that every five-dimensional Sasakian Lie algebra with trivial center is $\phi$-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 $-1$.