Related papers: Hermitian Structures on Lie Groups with Two-dimens…

This work is devoted to the study of a class of Poisson-Lie groups endowed with left invariant metrics. The triples $(G,\pi,<,>)$ are considered, where $G$ is a simply connected Lie group, ?$\pi$ is a multiplicative Poisson tensor and $<,>$…

Suppose we take an abelian group G and quotient it by the action of negation. What structure does the quotient K inherit from the group structure of G? We describe this structure (which we call the Kummer of G) in terms of a map from the…

In this work, we prove that, under a topological condition, the cohomology associated with left-invariant elliptic structures on compact semisimple Lie groups can be computed using only left-invariant forms. This reduces the analytical…

A Lie group as a 4-dimensional pseudo-Riemannian manifold is considered. This manifold is equipped with an almost product structure and a Killing metric in two ways. In the first case Riemannian almost product manifold with nonintegrable…

In this paper we apply the hyper-K\"ahler quotient construction to Lie groups with a left invariant hyper-K\"ahler structure under the action of a closed abelian subgroup by left multiplication. This is motivated by the fact that some known…

We give a general Lie-theoretic construction for anti-invariant almost Hermitian Riemannian submersions, anti-invariant quaternion Riemannian submersions, anti-invariant para-Hermitian Riemannian submersions, anti-invariant para-quaternion…

The characteristic connection of an almost hermitian structure is a hermitian connection with totally skew-symmetric torsion. The case of parallel torsion in dimension six is of particular interest. In this work, we give a full…

In the context of information geometry, the concept known as left-invariant statistical structure on Lie groups is defined by Furuhata--Inoguchi--Kobayashi (Inf Geom 4(1):177--188, 2021). In this paper, we introduce the notion of the moduli…

We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant…

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…

This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the…

Pseudo $H$-type Lie algebras are a special class of 2-step nilpotent metric Lie algebras, intimately related to Clifford algebras $\Cl_{r,s}$. In this work we propose the classification method for integral orthonormal structures of pseudo…

In the presented paper left-invariant pseudo-Riemannian metrics on four-dimensional Lie groups with zero Schouten-Weyl tensor are investigated. The complete classification of these metric Lie groups is obtained in terms of the structure…

In this paper, we introduce right-invariant Poisson-Nijenhuis Structures on Lie groupoids and their infinitesimal counterparts as called (Poisson bivector, Nijenhuis operator) structures. Also, we present a one-to-one correspondence between…

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 $H_{2n + 1}$.…

The definition and structure of hyperkahler structure preserving transformations (invariance group) for quaternionic structures have been recently studied and some preliminary results on the Euclidean case discussed. In this work we present…

The aim of this work is the study of left-invariant magnetic fields on 2-step nilpotent Lie groups. While the existence of closed 2-forms for which the center is either nondegenerate or in the kernel of the 2-form, is always guaranteed, the…

In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi-Civita connections and explicit formulas for…

This is an expanded version of a series of lectures delivered at the 25th Winter School ``Geometry and Physics'' in Srni. After a short introduction to Cartan geometries and parabolic geometries, we give a detailed description of the…

This paper is an adaptation of a chapter from an upcoming monograph on noncommutative geometry and quantum groups. We present examples of non compact quantum groups which are deformations of low dimensional Lie groups. The paper is of…