$η$-Einstein Sasakian Lie algebras
Adrián M. Andrada, Simon G. Chiossi, Alberth J. Nuñez
TL;DR
The paper develops a comprehensive framework for η-Einstein Sasakian structures on Lie algebras by separating centreful and centreless cases. In the centreful case, η-Einsteinness is equivalent to Kahler-Einstein-ness of the Kahler quotient, with unimodular examples necessarily null η-Einstein, and it provides explicit null solvmanifold constructions. For centreless algebras, a complete characterization is achieved under the assumption that Im adξ has dimension 2, reducing the problem to Kähler-exact algebras built from normal j-algebras and their modifications, enabling the construction of new η-Einstein Lie algebras and solvmanifolds and yielding finiteness and obstruction results. The work also shows that curvature is preserved under modifications, derives finiteness constraints on auxiliary data such as H0, and identifies an extreme semisimple boundary: in the maximal Im adξ case, centreless Sasakian algebras must be semisimple, specifically su(2) or sl(2,ℝ). Overall, these results provide new tools for constructing η-Einstein Sasakian Lie algebras and solvmanifolds and sharpen understanding of when such structures can exist.
Abstract
We study $η$-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 $η$-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 $η$-Einstein Sasakian Lie algebras and solvmanifolds, and provide effective restrictions for their existence.