Year: 2026
Type: article
Abstract: In this article, we provide a general set-up for arbitrary linear Lie groups $$H\le \textrm{GL}(n,\mathbb {R})$$ H ≤ GL ( n , R ) with Lie algebra $$\mathfrak {h}$$ 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 $$\mathfrak {g}=\mathfrak {g}_f$$ g = g f is fully determined by an endomorphism f of $$\mathbb {R}^{n-1}$$ R n - 1 , we give a description of the subspace $$\mathcal {F}_{\mathfrak {h}}$$ F h of all $$f\in {{\,\textrm{End}\,}}(\mathbb {R}^{n-1})$$ f ∈ End ( R n - 1 ) for which $$\mathfrak {g}_f$$ 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 $$\mathcal {F}_{\mathfrak {h}}$$ 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 , $$H\le \textrm{GL}(m,\mathbb {C})$$ 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 $$\mathcal {F}_{\mathfrak {h}}$$ F h coincides with what we call the characteristic subalgebra
Source: Journal of Geometric Analysis
Author: Marco Freibert
Institution: Universität Hamburg
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Funder: Universität Hamburg
OpenAlex ID: W7143658235