$\hat{I}\cdot$-Einstein Sasakian Lie algebras

Abstract

We study $\hat{I}\cdot$-Einstein Sasakian structures on Lie algebras, that is, Sasakian structures whose associated Ricci tensor satisfies an Einstein-like condition. We divide into the cases in which the Lie algebra's centre is non-trivial (and necessarily one-dimensional) and those where it is zero. In the former case we show that any Sasakian structure on a unimodular Lie algebra is $\hat{I}\cdot$-Einstein. As for centreless Sasakian Lie algebras, we devise a complete characterisation under certain dimensional assumptions regarding the action of the Reeb vector. Using this result, together with the theory of normal $j$-algebras and modifications of Hermitian Lie algebras, we construct new examples of $\hat{I}\cdot$-Einstein Sasakian Lie algebras and solvmanifolds, and provide effective restrictions for their existence.

34 pages. Newer version includes referee's comments, minor typographical changes, updated refs and acknowledgements. Accepted in Manuscripta Mathematica (Jan 2026)