In this article, we provide a general set-up for arbitrary linear Lie groups \(H\le \textrm{GL}(n,\mathbb {R})\) with Lie algebra \(\mathfrak {h}\) 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 \(\mathfrak {g}=\mathfrak {g}_f\) is fully determined by an endomorphism f of \(\mathbb {R}^{n-1}\) , we give a description of the subspace \(\mathcal {F}_{\mathfrak {h}}\) of all \(f\in {{\,\textrm{End}\,}}(\mathbb {R}^{n-1})\) for which \(\mathfrak {g}_f\) 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 \(\mathcal {F}_{\mathfrak {h}}\) 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 \(n=2m\) , \(H\le \textrm{GL}(m,\mathbb {C})\) 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 \(\mathcal {F}_{\mathfrak {h}}\) coincides with what we call the characteristic subalgebra \(\tilde{\mathfrak {k}}_{\mathfrak {h}}\) associated to \(\mathfrak {h}\) , and that then the torsion-free condition is equivalent to the left-invariant flatness condition. In particular, we prove this to be the case if H is a complex linear Lie group or if \(\mathfrak {h}\) does not contain any elements of rank one or two and is either metric or totally real.
