Torsion-Free H-Structures on Almost Abelian Solvmanifolds

Abstract In this article, we provide a general set-up for arbitrary linear Lie groups $$H\le \textrm{GL}(n,\mathbb {R})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>≤</mml:mo> <mml:mtext>GL</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with Lie algebra $$\mathfrak {h}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>h</mml:mi> </mml:math> 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 $$\mathfrak {g}=\mathfrak {g}_f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mrow> </mml:math> is fully determined by an endomorphism f of $$\mathbb {R}^{n-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> , we give a description of the subspace $$\mathcal {F}_{\mathfrak {h}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:math> of all $$f\in {{\,\textrm{End}\,}}(\mathbb {R}^{n-1})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mspace/> <mml:mtext>End</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for which $$\mathfrak {g}_f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:math> 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 $$\mathcal {F}_{\mathfrak {h}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:math> 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 $$n=2m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> , $$H\le \textrm{GL}(m,\mathbb {C})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>≤</mml:mo> <mml:mtext>GL</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$\mathcal {F}_{\mathfrak {h}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:math> coincides with what we call the characteristic subalgebra