Half-flat structures on decomposable Lie groups

TL;DR

This paper classifies all left-invariant half-flat SU(3)-structures on decomposable Lie groups, providing a complete solution to their existence problem through Lie algebra cohomology calculations.

Contribution

It offers a complete classification of half-flat SU(3)-structures on decomposable Lie groups, extending previous results and clarifying the structure of five-dimensional Lie algebras.

Findings

Complete classification of half-flat SU(3)-structures on decomposable Lie groups

Calculation of Lie algebra cohomology for all five-dimensional Lie algebras

Refinement of existing five-dimensional Lie algebra classification

Abstract

Half-flat SU(3)-structures are the natural initial values for Hitchin's evolution equations whose solutions define parallel G_2-structures. Together with the results of arXiv:0912.3486v1, the results of this article completely solve the existence problem of left-invariant half-flat SU(3)-structures on decomposable Lie groups. The proof is supported by the calculation of the Lie algebra cohomology for all indecomposable five-dimensional Lie algebras which refines and clarifies the existing classification of five-dimensional Lie algebras.