Flat left-invariant pseudo-Riemannian metrics on quadratic Lie groups

Abstract

A flat quadratic Lie algebra (g,〈,〉,k) is a Lie algebra g endowed with a flat pseudo-Euclidean metric 〈,〉 and a quadratic structure k. In geometrical terms, it is a Lie algebra of a Lie group endowed with both a flat left-invariant pseudo-Riemannian metric and a bi-invariant metric. It is known that if a Lie algebra admits a symplectic form and a quadratic structure then it is a flat quadratic Lie algebra. In this paper, we show that there exist non-abelian Lie algebras which admit a flat Lorentzian metric and a quadratic structure. If g is such Lie algebra, then we show that g is a trivial central extension of R⋊H2k+1 and [g,g]≃H2k+1, where H2k+1 is the Heisenberg Lie algebra and k≥2. In particular, we prove that g is 3-step solvable, not nilpotent and hence admits no symplectic structure. We show that there exists only one non-abelian Lie algebra of dimension n≤6 which admits a flat Lorentzian metric and a quadratic structure. Using a variant of the double extension method, which conserves both the flat and the quadratic metrics, we prove that any Lie algebra which admits a flat Lorentzian metric and a quadratic structure, is a double extension of an abelian Lie algebra and we construct a large class of such (non-abelian) Lie algebras in higher dimensions. Oscillator Lie algebras denoted by gλ constitute an important class of quadratic Lie algebras which are of the form R⋊H2k+1. It is known that gλ admits no flat Lorentzian metric. We show in this paper that gλ admits a Ricci-flat Lorentzian metric. The paper also study the class of quadratic 2-step nilpotent Lie algebras (g,k) which constitute examples of flat quadratic Lie algebras since k is also flat in this case.