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...

It is the aim of this work to study product structures on four dimensional solvable Lie algebras. We determine all possible paracomplex structures and consider the case when one of the subalgebras is an ideal. These results are applied to the case of Manin triples and complex product structures. We also analyze the three dimensional subalgebras.

A nilmanifold is a quotient of a nilpotent group G by a cocompact discrete subgroup. A complex nilmanifold is one which is equipped with a G-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of G-invariant complex structures which satisfy quaternionic relations). We prove that a hypercomplex nilmanifold admits an HKT (hyperkähler with torsion) metric if and only if the underlying hypercomplex structure is abelian. Moreover, any G-invariant HKT-metric on a nilmanifold is balanced with respect to all associated complex structures.

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Abstract. The purpose of this paper is to classify invariant hypercomplex structures on a 4-dimensional real Lie group G. It is shown that the 4-dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group H of the quaternions, the multiplicative group H * of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces, RH 4 and CH 2 , respectively, and the semidirect product C C. We show that the spaces CH 2 and C C possess an RP 2 of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian 4-manifolds are determined.

Abstract. In this paper we apply the hyper-Kähler quotient construction to Lie groups with a left invariant hyper-Kähler structure under the action of a closed abelian subgroup by left multiplication. This is motivated by the fact that some known hyper-Kähler metrics can be recovered in this way by considering different Lie group structures on H p × H q (H: the quaternions). We obtain new complete hyper-Kähler metrics on Euclidean spaces and give their local expressions.

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We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.2010 Mathematics Subject Classification. 53C15, 53B35, 53C30.