Homogeneous spaces, multi-moment maps and (2,3)-trivial algebras

TL;DR

This paper explores multi-moment maps in geometries with closed three-forms, focusing on homogeneous hypercomplex and nearly Kaehler manifolds, and introduces (2,3)-trivial Lie algebras with zero second and third Betti numbers.

Contribution

It provides concrete examples of multi-moment maps and classifies (2,3)-trivial Lie algebras up to dimension five, highlighting their role in geometric structures.

Findings

Examples of multi-moment maps for specific manifolds

Complete classification of (2,3)-trivial Lie algebras up to dimension five

Identification of the importance of (2,3)-trivial algebras in geometry

Abstract

For geometries with a closed three-form we briefly overview the notion of multi-moment maps. We then give concrete examples of multi-moment maps for homogeneous hypercomplex and nearly Kaehler manifolds. A special role in the theory is played by Lie algebras with second and third Betti numbers equal to zero. These we call (2,3)-trivial. We provide a number of examples of such algebras including a complete list in dimensions up to and including five.