Pseudo-Kähler and pseudo-Sasaki structures on Einstein solvmanifolds

Abstract

The aim of this paper is to construct left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups. We consider the class of $\mathfrak{z}$-standard Sasaki solvable Lie algebras of dimension $2n+3$, which are in one-to-one correspondence with pseudo-Kähler nilpotent Lie algebras of dimension $2n$ endowed with a compatible derivation, in a suitable sense. We characterize the pseudo-Kähler structures and derivations giving rise to Sasaki-Einstein metrics. We classify $\mathfrak{z}$-standard Sasaki solvable Lie algebras of dimension $\le 7$ and those whose pseudo-Kähler reduction is an abelian Lie algebra. The Einstein metrics we obtain are standard, but not of pseudo-Iwasawa type.

26 pages, 2 figures; v2: corrected Proposition 4.3 and Theorem 4.4 to account for the sign ambiguity; v3: improved notation and presentation; introduction and Section 1 expanded to illustrate the geometric meaning of the construction; Examples 3.9 and 4.6 expanded; title changed for clarity; added Example 2.6 and two figures

References in corpus (1)

• Pseudo-Riemannian Sasaki solvmanifolds

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