On the Riemann-Lie algebras and Riemann-Poisson Lie groups

It is shown that the Poisson structure related to $\kappa$-Poincar\'e group is dual to a certain Lie algebroid structure, the related Poisson structure on the (affine) Minkowski space is described in a geometric way.

We show that a Poisson Lie group $(G,\pi)$ is coboundary if and only if the natural action of $G\times G$ on $M=G$ is a Poisson action for an appropriate Poisson structure on $M$ (the structure turns out to be the well known $\pi _+$). We analyze the same condition in the context of Hopf algebras. Quantum analogue of the $\pi_+$ structure on SU(N) is described in terms of generators and relations as an example.

We investigate contact Lie groups having a left invariant Riemannian or pseudo-Riemannian metric with specific properties such as being bi-invariant, flat, negatively curved, Einstein, etc. We classify some of such contact Lie groups and derive some obstruction results to the existence of left invariant contact structures on Lie groups

This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie--Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearisability conditions, etc. Our results are illustrated by examples of physical and mathematical interest.

In recent years methods for the integration of Poisson manifolds and of Lie algebroids have been proposed, the latter being usually presented as a generalization of the former. In this note it is shown that the latter method is actually related to (and may be derived from) a particular case of the former if one regards dual of Lie algebroids as special Poisson manifolds. The core of the proof is the fact, discussed in the second part of this note, that coisotropic submanifold...

We define the concept of a flat pseudo-Riemannian $F$-Lie algebra and construct its corresponding double extension. This algebraic structure can be interpreted as the infinitesimal analogue of a Frobenius Lie group devoid of Euler vector fields. We show that the double extension provides a framework for generating all weakly flat Lorentzian non-abelian bi-nilpotent $F$-Lie algebras possessing one dimensional light-cone subspaces. A similar result can be established for nilpot...

These lecture notes in Lie Groups are designed for a 1--semester third year or graduate course in mathematics, physics, engineering, chemistry or biology. This landmark theory of the 20th Century mathematics and physics gives a rigorous foundation to modern dynamics, as well as field and gauge theories in physics, engineering and biomechanics. We give both physical and medical examples of Lie groups. The only necessary background for comprehensive reading of these notes are a...

We consider some special type extensions of an arbitrary Lie algebra ${\cal G}$, arising in the theory of Lie-Poisson structures over $({\cal G}^*)^n$, where ${\cal G}^*$ is the dual of ${\cal G}$. We show that some classes of these extensions can be constructed in a natural way using some linear bundles of Lie algebras.

Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie algebras and compact Lie algebras we recover by simple methods interesting co-homological known results. The Lie algebra of prederivations encompasses the on...

In this paper, we mainly study left invariant pseudo-Riemannian Ricci-parallel metrics on connected Lie groups which are not Einstein. Following a result of Boubel and B\'{e}rard Bergery, there are two typical types of such metrics, which are characterized by the minimal polynomial of the Ricci operator. Namely, its form is either $(X-\alpha)(X-\bar{\alpha})$ (type I), where $\alpha\in \mathbb{C}\setminus \mathbb{R}$, or $X^{2}$ (type II). Firstly, we obtain a complete descri...