Half-flat structures on indecomposable Lie groups

Abstract

This article can be viewed as a continuation of the articles [12] and [5] where the decomposable Lie algebras admitting half-flat SU(3)-structures are classified. The new main result is the classification of the indecomposable six-dimensional Lie algebras with five-dimensional nilradical which admit a half-flat SU(3)-structure. As an important step of the proof, a considerable refinement of the classification of six-dimensional Lie algebras with five-dimensional non-Abelian nilradical is established. Additionally, it is proved that all non-solvable six-dimensional Lie algebras admit half-flat SU(3)-structures.

Six-Dimensional Solvmanifolds with Holomorphically Trivial Canonical Bundle

“…, e 6 } is the basis of g * in which the Lie algebra is expressed. Notice that we will follow the notation given in [14,15,28] and [31] to name the Lie algebras; for instance, the notation e(2) ⊕ e(1, 1) = (0, −e 13 , e 12 , 0, −e 46 , −e 45 ) means that e(2) ⊕ e(1, 1) is the (decomposable) Lie algebra determined by a basis {e i } 6 i=1 such that de 1 = 0, de 2 = −e 13 , de 3 = e 12 , de 4 = 0, de 5 = −e 46 , de 6 = −e 45 . The next two concrete examples show how we will proceed in general in the proofs of Propositions 2.6 and 2.7 below in order to exclude candidates.…”

“…comes from the classification in [31], and they are the only unimodular solvable Lie algebras with b 3 ≥ 2 and nilradical of dimension 4. The other Lie algebras in Table 2 are taken from [15]. In Table 2 we also include the column "λ(ρ) ≥ 0" in which the symbol means that any closed 3-form ρ on the Lie algebra satisfies λ(ρ) ≥ 0, in particular, ρ does not give rise to an almost complex structure (a similar study was done in [14] for any decomposable Lie algebra).…”

Cohomological properties of unimodular six dimensional solvable Lie algebras

“…Next we list 6-dimensional unimodular, solvable, non-nilpotent Lie algebra g together with their first, second and third Betti number. The Betti numbers of the 6-dimensional Lie algebras with 5-dimensional nilradical were also computed by M. Freibert and F. F. Schulte-Hengesbach [4].…”

“…The Betti numbers of the 6-dimensional Lie algebras with 5-dimensional nilradical were also computed by M. Freibert and F. F. Schulte-Hengesbach [4].…”

Half-flat structures inducing Einstein metrics on homogeneous spaces

“…These classifications will be useful in the proof of the following Proof. We will prove the theorem as follows: in the list of Einstein solvable metric Lie algebras we first exclude the ones that do not admit a half-flat structure using the results of [17] and [18], then we will show the result by direct computations in the remaining cases.…”