Half-flat structures on decomposable lie groups

Half-at 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 [SH], the results of this article completely solve the existence problem of left-invariant half-at 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.