A. Andrada

Classification of abelian complex structures on 6-dimensional Lie algebras

We classify the 6-dimensional Lie algebras that can be endowed with an abelian com- plex structure and parameterize, on each of these algebras, the space of such structures up to holo- morphic isomorphism.

Double Products and Hypersymplectic Structures on ℝ4n

In this paper we give a procedure to construct hypersymplectic structures on ℝ4n beginning with affine-symplectic data on ℝ2n. These structures are shown to be invariant by a 3-step nilpotent double Lie group and the resulting metrics are complete and not necessarily flat. Explicit examples of this construction are exhibited.

Corrigendum: Classification of abelian complex structures on six-dimensional Lie algebras

It has been pointed to us by E. Rodŕıguez Valencia that the complex structures J t and J 2 t on the Lie algebra n4, appearing in Theorem 3.3, are in fact equivalent. These structures are introduced in the proof of Theorem 3.2, which is used later in the paper to determine the moduli spaces of abelian complex structures. However, in that proof, statement (J) is in fact impossible, since it implies that the Lie algebra g is abelian. Indeed, if ker(adx |v) were J-stable for any x ∈ v, then we would have [x, Jx] = 0 for any x ∈ g. Therefore, for any x, y ∈ g,

Locally conformally Kähler structures on unimodular Lie groups

We study left-invariant locally conformally Kähler structures on Lie groups, or equivalently, on Lie algebras. We give some properties of these structures in general, and then we consider the special cases when its complex structure is bi-invariant or abelian. In the former case, we show that no such Lie algebra is unimodular, while in the latter, we prove that if the Lie algebra is unimodular, then it is isomorphic to the product of

Product structures on four dimensional solvable Lie algebras

\mathbb {R}

Hypersymplectic Lie algebras

R and a Heisenberg Lie algebra.

Abelian Hermitian geometry

Given an almost complex manifold (M, J), we study complex connections with trivial holonomy such that the corresponding torsion is either of type (2, 0) or of type (1, 1) with respect to J. Such connections arise naturally when considering Lie groups, and quotients by discrete subgroups, equipped with bi-invariant and abelian complex structures.