Torsion-Free H-Structures on Almost Abelian Solvmanifolds

Abstract In this article, we provide a general set-up for arbitrary linear Lie groups H GL (n, R) H ≤ GL (n, R) with Lie algebra h h which allows to characterise the almost Abelian Lie algebras admitting a torsion-free H -structure. In more concrete terms, using that an n -dimensional almost Abelian Lie algebra g= gf g = g f is fully determined by an endomorphism f of R^n-1 R n - 1, we give a description of the subspace F ₇ F h of all f {\, End\, } (R^n-1) f ∈ End (R n - 1) for which gf g f admits a “special” torsion-free H -structure in terms of the image of a certain linear map. For large classes of linear Lie groups H, we are able to explicitly compute F ₇ F h and so give characterisations of the almost Abelian Lie algebras admitting a torsion-free H -structure. Our results reprove all the known characterisations of the almost Abelian Lie algebras admitting a torsion-free H -structure for different single linear Lie groups H and extends them to big classes of linear Lie groups H. For example, we are able to provide characterisations in the case n=2m n = 2 m, H GL (m, C) H ≤ GL (m, C) and H either being a complex Lie group or being totally real, or in the case that H preserves a pseudo-Riemannian metric. In many cases, we show that the space F ₇ F h coincides with what we call the characteristic subalgebra