The structure of underlying Lie algebras

Abstract

Every Lie algebra over a field E gives rise to new Lie algebras over any subfield F ⊆ E by restricting the scalar multiplication. This paper studies the structure of these underlying Lie algebra in relation to the structure of the original Lie algebra, in particular the question how much of the original Lie algebra can be recovered from its underlying Lie algebra over subfields F . By introducing the conjugate of a Lie algebra we show that in some specific cases the Lie algebra is completely determined by its underlying Lie algebra. Furthermore we construct examples showing that these assumptions are necessary.As an application, we give for every positive n an example of a real 2-step nilpotent Lie algebra which has exactly n different bi-invariant complex structures. This answers an open question by Di Scala, Lauret and Vezzoni motivated by their work on quasi-Kähler Chern-flat manifolds in differential geometry. The methods we develop work for general Lie algebras and for general Galois extensions F ⊆ E, in contrast to the original question which only considered nilpotent Lie algebras of nilpotency class 2 and the field extension R ⊆ C. We demonstrate this increased generality by characterizing the complex Lie algebras of dimension ≤ 4 which are defined over R and over Q.

Orthogonal bi-invariant complex structures on metric Lie algebras

“…But also the case F = C is crucial, because of its relation to bi-invariant complex structures on the underlying real Lie algebra. Hence we formulate some of the results in this paper, in particular Theorem 3.1, for general subfields of C, comparable to [6].…”

“…] shows that the indecomposable factors of a decomposition are unique up to isomorphism. However, [6,Example 2.3. ] demonstrates that these isomorphic factors can be different ideals of the Lie algebra and thus two decompositions are not necessarily identical.…”

“…In the special case of irreducible Lie algebras which are 2-step nilpotent, Theorem 4.2 was given in [3,Proposition 4.9.]. For Lie algebras without inner product, [6] shows that any Lie algebra has at most finitely many bi-invariant complex structures up to isomorphism and gives an explicit way to describe all of them, similarly as in Theorem 4.2. The main tool in that proof was to consider so-called conjugate Lie algebras for elements of the Galois group of a field extension.…”

“…In this way, Question 1 can be considered as the metric counterpart of [7,Question 5], which asked whether two different complex Lie algebras could have an isomorphic underlying real Lie algebra. Examples of such complex Lie algebras, including a general way of describing them, was provided in [6].…”

Almost inner derivations of Lie algebras

“…Let k ⊆ K be a Galois extension and g K a Lie algebra over K with underlying Lie algebra g ′ k . A more general result about the structure of g ′ K can be found in [3]. Now, we return again to the general situation where [K : k] = n ≥ 2.…”