Classification of abelian complex structures on 6-dimensional Lie algebras

Abstract

We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parametrize, on each of these algebras, the space of such structures up to holomorphic isomorphism. IntroductionLet g be a Lie algebra, J be an endomorphism of g such that J 2 = −I, and let g 1,0 be the i-eigenspace of J in g C := g ⊗ R C. When g 1,0 is a complex subalgebra, we say that J is integrable; when g 1,0 is abelian, we say that J is abelian; and when g 1,0 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 bi-invariant, 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 [17]). However, its nilradical need not be abelian (see Remark 8). Abelian complex structures have interesting applications in hyper-Kähler with torsion geometry (see [6]). It has been shown in [9] 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 [10].Of importance, when studying complex structures on a Lie algebra g, is the ideal g J := g + Jg constructed from algebraic and complex data. We say that the complex structure J is proper when g J is properly contained in g. Any complex structure on a nilpotent Lie algebra is proper [19]. The 6-dimensional nilpotent Lie algebras carrying complex structures were classified in [19], and those carrying abelian complex structures were classified in [12].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 [20]. 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 2-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. Mathematics Subject Classification 17B30 (primary), 53C15 (secondary).The authors were partially supported by CONICET, ANPCyT and SECyT-UNC (Argentina). n 2 . Furthermore, for 0 r, s n 2 , if J r is equivalent to J s , by comparin...

Chern-flat and Ricci-flat invariant almost Hermitian structures

“…Remark 3. 4 We do not know if Corollary 1.3 is true when G is not nilpotent, e.g. for solvmanifolds.…”

“…Abelian complex structures were introduced in [5] and were intensely studied in [4,6,11,8,19]. Observe that J is anti-abelian and integrable if and only if it is bi-invariant, i.e.…”

Invariants of complex structures on nilmanifolds

“…Barberis and I.G. Dotti in [1] gave a classification of all 6-dimensional Lie algebras admitting an abelian complex structure; furthermore, they give a parametrization, on each Lie algebra, of the space of abelian structures up to holomorphic isomorphism. In particular, there are three nilpotent Lie algebras carrying curves of non-equivalent structures.…”

“…Based on this parametrization, we study the existence of minimal metrics on each of these complex nilmanifolds (see Theorem 4.4), and provide an alternative proof of the pairwise non-isomorphism between the structures. The classification in [1] fix the Lie algebra and varies the complex structure. For example, on the Lie algebra h 3 × h 3 they found the curve J s of abelian complex structures defined by J s e 1 = e 2 , J s e 3 = e 4 , J s e 5 = se 5 + e 6 , s ∈ R, and fix the bracket [e 1 , e 2 ] = e 5 , [e 3 , e 4 ] = e 6 .…”

“…We now will use the Pfaffian forms to give an alternative proof of the pairwise non-isomorphism of the family given in [1,Theorem 3.4] in the 2-step nilpotent case. Since dim v 1 = 4 and dim v 2 = 2 , the Pfaffian forms of n 1 , .…”