Invariant metrics on central extensions of quadratic Lie algebras

Abstract

A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and nondegenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central extensions of quadratic Lie algebras which in turn have invariant metrics. The structure is such that the central extensions can be described algebraically in terms of the original quadratic Lie algebra, and geometrically in terms of the direct sum decompositions that the invariant metrics involved give rise to.

Hom-Lie algebra structures on quadratic Lie algebras and twisted invariant Killing-like forms defined on them

“…Observe that (7) states that α ′ ∈ Γ B (g) (see Prop. 2.4 of [5] for a proof). On the other hand, since µ(…”

“…All Lie algebras in this paper are finite dimensional over an algebraically closed field F of characteristic zero. We shall closely follow the approach of [5] with a slight change of notation. For the sake of being self-contained and for our readers' benefit, we reproduce here the setting used in [5] and quote the main results there on which we now base this work.…”

“…. Observe that no a priori relationship between B 0 and B is assumed (eg, B 0 is not necessarily the restriction of B to g 0 ) and therefore g 0 is neither a subalgebra nor an ideal of g. Nevertheless, the linear maps h : g → g 0 and k : g 0 → g, help each other to determine B in terms of B 0 and viceversa, as the following result from [5] states: 1.3. Lemma.…”

“…But the important point for us is that ρ somehow intertwines the geometry and the algebra between g and g 0 . In order to explain how, we first quote the following result from [5] (see Lemma 2.2 therein) that recalls some of the properties of ρ.…”

“…The origin of this work is [5], where the following problem is addressed and solved: Given a quadratic Lie algebra g having a subspace (not necessarily a subalgebra) g 0 ֒→ g that admits the structure of a quadratic Lie algebra whose invariant bilinear form B 0 : g 0 × g 0 → F is not given by the restriction of the invariant bilinear form B : g × g → F, could it carry an ideal V ֒→ g in such a way that g is an extension of g 0 ? We have shown in [5] how to reconstruct g from g 0 and V and how to produce an invariant B on it, given the fact that extension V ֒→ g → g/V ≃ g 0 is central. We shall refer ourselves to such central extensions g equipped with their corresponding invariant B, as quadratic central extensions for short.…”