arXiv research

Study of Riemann-Poisson Lie groups with compatibility conditions.

A $n$-dimensional Lie group $G$ equipped with a left invariant symplectic form $\om^+$ is called a symplectic Lie group. It is well-known that $\om^+$ induces a left invariant affine structure on $G$. Relatively to this affine structure we show that the left invariant Poisson tensor $π^+$ corresponding to $\om^+$ is po…

This work is devoted to the study of a class of Poisson-Lie groups endowed with left invariant metrics. The triples $(G,π,<,>)$ are considered, where $G$ is a simply connected Lie group, ?$π$ is a multiplicative Poisson tensor and $<,>$ is a left invariant riemannian metric such that Hawkins conditions are satisfied. H…

Let G be a Lie group with Lie algebra $ \Cal G: = T_εG$ and $T^*G = \Cal G^* \rtimes G$ its cotangent bundle considered as a Lie group, where G acts on $\Cal G^*$ via the coadjoint action. We show that there is a 1-1 correspondance between the skew-symmetric solutions $r\in \wedge^2 \Cal G$ of the Classical Yang-Baxter…

We study the triple $(G,π,\prs)$ where $G$ is a connected and simply connected Lie group, $π$ and $\prs$ are, respectively, a multiplicative Poisson tensor and a left invariant Riemannian metric on $G$ such that the necessary conditions, introduced by Hawkins, to the existence of a non commutative deformation (in the d…

Characterizes symmetric Killing tensors on specific Lie groups.

We describe the generalized Kuranishi spaces of solvmanifolds with left-invariant complex structures. By using such description, we study the stability of left-invariantness of deformed generalized complex structures and smoothness of generalized Kuranishi spaces on certain classes of solvmanifolds. We also give explic…

We study left-invariant symmetric Killing 2-tensors on 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric, and construct genuine examples, which are not linear combinations of parallel tensors and symmetric products of Killing vector fields.

In this paper we study left invariant CR structures on Lie groups which are compatible with geometric properties as Poisson and kahler properties.

The paper explores geometric and algebraic structures on Lie groups.

Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.

A Riemann-Lie algebra is a Lie algebra $\cal G$ such that its dual ${\cal G}^*$ carries a Riemannian metric compatible (in the sense introduced by th author in C. R. Acad. Paris, t. 333, Série I, (2001) 763-768) with the canonical linear Poisson sructure of ${\cal G}^*$. The notion of Riemann-Lie algebra has its origin…

Study on 3D Lie groups finds all generalized Einstein metrics.

The paper studies geometric properties of tangent Poisson-Lie groups.

The main purpose of this paper is to investigate the Schouten-Weyl tensor on the three-dimensional Lie groups with left-invariant Lorenzian metrics. The left-invariant Lorentzian metrics on the three-dimensional Lie groups with squared length zero Schouten-Weyl tensor are studied. Moreover, the three-dimensional metric…

New findings on Codazzi tensors in homogeneous spaces.

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 constants of corresponding Lie algebras.

The paper defines and explores Poisson-Nijenhuis structures on Lie groupoids.

Study odd generalized Einstein metrics on 3D Lie groups.

We call a metric $m$-quasi-Einstein if $Ric_X^m$ (a modification of the $m$-Bakry-Emery Ricci tensor in terms of a suitable vector field $X$) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant vector fields and…

Let $G$ be a Lie group of even dimension and let $(g,J)$ be a left invariant anti-Kähler structure on $G$. In this article we study anti-Kähler structures considering the distinguished cases where the complex structure $J$ is abelian or bi-invariant. We find that if $G$ admits a left invariant anti-Kähler structure $(g…

In the present paper, we study complete and vertical lifts of tensor fields from a smooth manifold $M$ to its Weil bundle $T^A M$ defined by a Frobenius Weil algebra $A$. For a Poisson manifold $(M,w)$, we show that the complete lift $w^C$ and the vertical lift $w^V$ of the Poisson tensor $w$ are Poisson tensors on $T^…

A unified framework for Poisson and Jacobi structures from 2-covariant tensors

In this paper we prove that both complete and vertical lifts of a Poisson vector field from a Poisson manifold $(M, π)$ to its tangent bundle $(TM, π_{TM})$ are also Poisson. We use this fact to describe the infinitesimal deformations of Poisson tensor $π_{TM}$. We study some of their properties and present a extensive…

The paper classifies metrics on Heisenberg group's cotangent bundle.

The paper constructs a Poisson algebra bundle for multilocal observables.

The aim of this paper is to determine left-invariant strictly almost Kähler structures on 4-dimensional Lie groups $(g, J, Ω)$ such that the Ricci tensor is $J$-invariant.

New generalized Poisson structures are introduced by using suitable skew-symmetric contravariant tensors of even order. The corresponding `Jacobi identities' are provided by conditions on these tensors, which may be understood as cocycle conditions. As an example, we provide the linear generalized Poisson structures wh…

Study invariant CKY 2-forms on 5D Lie groups, classifying and determining their properties.

Proposes a method to handle sparse multiway count data with false zeros using zero-truncated Poisson regression.

The paper explores the connection between Poisson-Lie structures and invariant volume forms in Hamiltonian dynamics.

The paper studies Riemannian metrics on tangent Lie groups using two left-invariant metrics.

We study invariant Nijenhuis $(1,1)$-tensors on a homogeneous space $G/K$ of a reductive Lie group $G$ from the point of view of integrability of a Hamiltonian system of differential equations with the $G$-invariant Hamiltonian function on the cotangent bundle $T^*(G/K)$. Such a tensor induces an invariant Poisson tens…

The paper constructs compatible Poisson brackets on gl(N).

Newly introduced generalized Poisson structures based on suitable skew-symmetric contravariant tensors of even order are discussed in terms of the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are expressed as conditions on these tensors, the cohomological contents of which is given. In particular, we …

We show how to reduce, under certain regularities conditions, a Poisson-Nijenhuis Lie algebroid to a symplectic-Nijenhuis Lie algebroid with nondegenerate Nijenhuis tensor. We generalize the work done by Magri and Morosi for the reduction of Poisson-Nijenhuis manifolds. The choice of the more general framework of Lie a…

We prove unobstructed deformations for compact Kaehlerian even-dimensional Poisson manifolds whose Poisson tensor degenerates along a divisor with mild singularities. Examples include Hilbert schemes of del Pezzo surfaces.

Study the rank of Nijenhuis tensor on parallelizable almost complex manifolds.

We formulate and solve a tensor model using a latent-variable approach.

Let M be a Poisson manifold and A a Weil algebra. We describe an isomorphism of cohomolgy algebra and proves that Poisson cohomology with values in A is isomorphic to the tensor product of A with Poisson cohomolgy with real values.

The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.

The paper finds solutions for specific curvature conditions on 5D Lie groups.

Study of equivariant Poisson 2-algebra bundles over configuration spaces.

We study a class of Poisson-Nijenhuis systems defined on compact hermitian symmetric spaces, where the Nijenhuis tensor is defined as the composition of Kirillov-Konstant-Souriau symplectic form with the so called Bruhat-Poisson structure. We determine its spectrum. In the case of Grassmannians the eigenvalues are the …

Study of complex and Hermitian structures on specific Lie groups.

Small Nijenhuis tensor found on compact manifolds.

Poisson-NIjenhuis structures for an arbitrary Lie agebroid are defined and studied by means of tangent lifts of tensor fields.

Study Lie bialgebra structures on flat metric Lie algebras, leading to explicit Poisson-Lie groups.