Related papers: Left invariant special K\"ahler structures

We present two constructions of complex symplectic structures on Lie algebras with large abelian ideals. In particular, we completely classify complex symplectic structures on almost abelian Lie algebras. By considering compact quotients of…

In these notes we study left-invariant involutive structures on $\mathrm{SU}(2)$, the most na\"ive non-commutative compact Lie group. We determine closedness of the range (in the smooth topology) of a single complex vector field spanning…

We study existence of complex structures on semidirect products $\g \oplus_{\rho} \v$ where $\g$ is a real Lie algebra and $\rho$ is a representation of $\g$ on $\v$. Our first examples, the Euclidean algebra $\e(3)$ and the Poincar\'e…

In this paper the classification of left-invariant Riemannian metrics, up to the action of the automorphism group, on cotangent bundle of (2n+1)-dimensional Heisenberg group is presented. Also, it is proved that the complex structure on…

Let G be a Lie group, $T^*G$ its cotangent bundle with its natural Lie group structure obtained by performing a left trivialization of T^*G and endowing the resulting trivial bundle with the semi-direct product, using the coadjoint action…

In 1976, Milnor classified all Lie groups admitting a flat left-invariant metric. They form a special type of unimodular 2-step solvable groups. Considering Lie groups with Hermitian structure, namely, a left-invariant complex structure and…

In the present paper, we describe two geometric notions, holomorphic Norden structures and K\"{a}hler-Norden structures on Hom-Lie groups, and prove that on Hom-Lie groups in the left invariant setting, these structures are related to each…

We construct local models of isolated singularities for special K\"ahler structures in real dimension two assuming that the associated holomorphic cubic form does not have essential singularities. As an application we compute the holonomy…

We study 4-dimensional simply connected Lie groups $G$ with left-invariant Riemannian metric $g$ admitting non-trivial conformal Killing 2-forms. We show that either the real line defined by such a form is invariant under the group action,…

In these notes we survey basic concepts of affine geometry and their interaction with Riemannian geometry. We give a characterization of affine manifolds which has as counterpart those pseudo-Riemannian manifolds whose Levi-Civita…

In this paper we briefly survey the classical problem of understanding which Lie algebras admit a complex structure, put in the broader perspective of almost complex structures with special properties. We focus on the different behavior of…

In this paper we give a procedure to construct hypersymplectic structures on $R^{4n}$ beginning with affine-symplectic data on $R^{2n}$. These structures are shown to be invariant by a 3-step nilpotent double Lie group and the resulting…

Inspired by the work of Chevalley and Eilenberg on the de Rham cohomology on compact Lie groups, we prove that, under certain algebraic and topological conditions, the cohomology associated to left-invariant elliptic, and even hypocomplex,…

We find three-dimensional subspaces of four-dimensional connected Lie algebras, generating these algebras, and abnormal extremals on connected Lie groups with these Lie algebras and with left-invariant sub-Finsler quasimetrics defined by…

We construct a model space $C(\gsp(\bR^{2n}))$ for the variety of Abelian simply transitive groups of affine transformations of type ${\rm Sp}(\bR^{2n})$. The model is stratified and its principal stratum is a Zariski-open subbundle of a…

We characterise Lie groups with bi-invariant bargmannian, galilean or carrollian structures. Localising at the identity, we show that Lie algebras with ad-invariant bargmannian, carrollian or galilean structures are actually determined by…

We consider the 2-generated free metabelian associative and Lie algebras over the complex field and the invariants of the dihedral groups of finite order acting on these algebras. In the associative case we find a finite set of generators…

Each choice of a K\"ahler class on a compact complex manifold defines an action of the Lie algebra $\slt$ on its total complex cohomology. If a nonempty set of such K\"ahler classes is given, then we prove that the corresponding…

Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra, called big algebra, attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are…

In this paper we consider left-invariant pseudo-K\"{a}hler structures on six-dimensional nilpotent Lie algebras. The explicit expressions of the canonical complex structures are calculated, and the curvature properties of the associated…