Classification of abelian complex structures on 6-dimensional Lie algebras

Let g be a Lie algebra, J an endomorphism of g such that J = −I , and let g be the ieigenspace of J in g := g ⊗R C. When g is a complex subalgebra we say that J is integrable, when g is abelian we say that J is abelian and when g is a complex ideal we say that J is bi-invariant. We note that a complex structure on a Lie algebra cannot be both abelian and biinvariant, unless the Lie bracket is trivial. If G is a connected Lie group with Lie algebra g, by left translating J one obtains a complex manifold (G, J) such that left multiplication is holomorphic and, in the bi-invariant case, also right multiplication is holomorphic, which implies that (G, J) is a complex Lie group. Our concern here will be the case when J is abelian. In this case the Lie algebra has abelian commutator, thus, it is 2-step solvable (see [16]). However, its nilradical need not be abelian (see Remark 4.7). Abelian complex structures have interesting applications in hyper-Kähler with torsion geometry (see [6]). It has been shown in [8] that the Dolbeault cohomology of a nilmanifold with an abelian complex structure can be computed algebraically. Also, deformations of abelian complex structures on nilmanifolds have been studied in [9]. Of importance, when studying complex structures on a Lie algebra g, is the ideal gJ := g ′ + Jg constructed from algebraic and complex data. We will say that the complex structure J is proper when gJ is properly contained in g. Any complex structure on a nilpotent Lie algebra is proper [18]. The classification of nilpotent 6-dimensional Lie algebras carrying complex structures has been given in [18]; when the complex structure is abelian the characterization was obtained in [11]. There is only one 2-dimensional non-abelian Lie algebra, the Lie algebra of the affine motion group of R, denoted by aff(R). It carries a unique complex structure, up to equivalence, which turns out to be abelian. The 4-dimensional Lie algebras admitting abelian complex structures were classified in [19]. Each of these Lie algebras, with the exception of aff(C), the realification of the Lie algebra of the affine motion group of C, has a unique abelian complex structure up to equivalence. On aff(C) there is a two-sphere of abelian complex structures, but only two equivalence classes distinguished by J being proper or not. Furthermore, aff(C) is equipped with a natural bi-invariant complex structure. In dimension 6 it turns out that, as a consequence of our results, some of the Lie algebras equipped with abelian complex structures are of the form aff(A) where A is a 3-dimensional commutative associative algebra.