A study is made of real Lie algebras admitting compatible complex and product structures, including numerous 4-dimensional examples. Any Lie algebra g with such a structure is even-dimensional and its complexification has a hypercomplex structure. In addition, g splits into the direct sum of two Lie subalgebras of the same dimension, and each of these is shown to have a left-symmetric algebra (LSA) structure. Interpretations of these results are obtained that are relevant to the theory of both hypercomplex and hypersymplectic manifolds and their associated connections.

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.

We study the existence of lattices in almost abelian Lie groups that admit left invariant locally conformal Kähler or locally conformal symplectic structures in order to obtain compact solvmanifolds equipped with these geometric structures. In the former case, we show that such lattices exist only in dimension 4, while in the latter case we provide examples of such Lie groups admitting lattices in any even dimension.

We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group H 2n+1 . Furthermore, we classify Sasakian Lie algebras of dimension five and determine which of them carries a Sasakian α-Einstein structure. We show that a five-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either H 5 or a semidirect product R (H 3 × R). In particular, the compact quotient is an S 1 -bundle over a four-dimensional Kähler solvmanifold.