Torsion-Free H-Structures on Almost Abelian Solvmanifolds

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 Rⁿ⁻¹ R n - 1 , we give a description of the subspace Fₕ F h of all f∈ \,End\,(Rⁿ⁻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 Fₕ 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 Fₕ F h coincides with what we call the characteristic subalgebra