686 Results Found

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a compl...

This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th...

Let G be a Poisson–Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on G to the tangent bundle $TG$ of $G.$ In this paper, we induce a left invariant contravariant ps...

We review Poisson–Lie groups and their applications in gauge theory and integrable systems from a mathematical physics perspective. We also comment on recent results and developments and their applications. In particular, we discuss the role of quasi...

We give new concise Lie-theoretic proofs of basic analytic–geometric properties of connected complex Lie groups. Using Matsushima’s biholomorphic splitting $G\simeq {\mathbb{C}}^n\times \tilde{K}$ together with a refined analysis of the center via its C...

A statistical transformation model consists of a smooth data manifold, on which a Lie group smoothly acts, together with a family of probability density functions on the data manifold parametrized by elements in the Lie group. For such a statistical...

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space...

In computational anatomy, organ’s shapes are often modeled as deformations of a reference shape, i.e., as elements of a Lie group. To analyze the variability of the human anatomy in this framework, we need to perform statistics on Lie groups. A Lie g...

Given a compact connected Lie group G with an $S^1$ -module structure and a maximal compact torus T of $G^{S^1}$ , we define twisted Weyl group $W(G,S^1,T)$ of G associated to $S^1$ -module and show that two elements of...

In this paper, we address the problem of estimating the position of a mobile such as a drone from noisy position measurements using the framework of Lie groups. To model the motion of a rigid body, the relevant Lie group happens to be the Special Euc...

This is mainly a review of an intense 15-year long collaboration between the authors on explicit realisations of compact Lie groups and their applications. Starting with an elementary example, we will illustrate the main idea at the foundation of the...

We introduce poly-symplectic extension of Souriau Lie groups thermodynamics based on higher-order model of statistical physics introduced by Ingarden. This extended model could be used for small data analytics and machine learning on Lie groups. Sour...

We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the method of nonco...

Neural ordinary differential equations (neural ODEs) are a well-established tool for optimizing the parameters of dynamical systems, with applications in image classification, optimal control, and physics learning. Although dynamical systems of inter...

Thestability of a control system is essential for its effective operation. Stability implies that small changes in input, initial conditions, or parameters do not lead to significant fluctuations in output. Various stability properties, such as inner...

We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators o...

I discuss old and new results on fixed points of local actions by Lie groups G on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed.

This work investigates the stability analysis of linear control systems defined on Lie groups, with a particular focus on low-dimensional cases. Unlike their Euclidean counterparts, such systems evolve on manifolds with non-Euclidean geometry, where...

In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, $H\subset K\subset G$, such that $G/K$ is a compact, connected, irreducible, symmetric space, and the isotropy representation of $G/H$...

Riemann solitons are generalized fixed points of the Riemann flow. In this note, we study left-invariant Riemann solitons on three-dimensional Lorentzian Lie groups. We completely classify left-invariant Riemann solitons on three-dimensional Lorentzi...

In this paper, we study left-invariant cross curvature solitons on Lorentzian three-dimensional Lie groups and classify these solitons.

We consider the phenomenon of concentration of measures, which is restricted to the case of families of compact connected Lie groups. While in the literature, powerful general results regarding the existence of concentration and its relations to extr...

In 2016, Wears defined and studied algebraic T-solitons. In this paper, we define algebraic Schouten solitons as a special T-soliton and classify the algebraic Schouten solitons associated with Levi-Civita connections, canonical connections, and Koba...

We investigate left-invariant generalized cross-curvature solitons on simply connected three-dimensional Lorentzian Lie groups. Working with the assumption that the contravariant tensor $P_{ij}$ (defined from the Ricci tensor and scalar curvature) is inve...

The present paper recalls a formulation of non-conservative system dynamics through the Lagrange–d’Alembert principle expressed through a generalized Euler–Poincaré form of the system equation on a Lie group. The paper illust...

In this paper, we study contact magnetic geodesics in a 3-dimensional Lie group G endowed with a left invariant almost cosymplectic structure. We distinguish the two cases: G is unimodular, and G is nonunimodular. We pay a careful attention to the sp...

In total, geodesic surfaces and their generalizations, namely totally umbilical and parallel surfaces, are well-known topics in Submanifold Theory and have been intensively studied in three-dimensional ambient spaces, both Riemannian and Lorentzian....

In this paper, we derive the expressions of Codazzi tensors associated with Yano connections in seven Lorentzian Lie groups. Furthermore, we complete the classification of three-dimensional Lorentzian Lie groups in which Ricci tensors associated with...

This review article explores the theory of control sets for linear control systems defined on two-dimensional Lie groups, with a focus on the plane ${\mathbb{R}}^2$ and the affine group $Af{f}_{+}\left(2\right)$. We systematically summarize recent advances, emphasizing how the geom...

We studied the random variable $V_t={\mathrm{vol}}_{S^2}(g_{t}B\cap B)$, where B is a disc on the sphere $S^2$ centered at the north pole and ${\left(g_t\right)}_{t\ge 0}$ is the Brownian motion on the special orthogonal group $SO\left(3\right)$ starting at the identity. We applied the results of the theo...

Motivated by the rapid progress of aerospace and robotics engineering, the navigation and control systems on matrix Lie groups have been actively studied in recent years. For rigid targets, the attitude estimation problem is a benchmark one with its...

We present schemes for simulating Brownian bridges on complete and connected Lie groups and homogeneous spaces. We use this to construct an estimation scheme for recovering an unknown left- or right-invariant Riemannian metric on the Lie group from s...

Hyperspectral imagery (HSI) offers rich spectral signatures and fine-grained spatial structures for remote sensing, but practical HSI classification is often constrained by scarce labels and complex geometric disturbances, including translation, rota...

We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physi...

Attitude estimation is a basic task for most spacecraft missions in aerospace engineering and many Kalman type attitude estimators have been applied to the guidance and navigation of spacecraft systems. By building the attitude dynamics on matrix Lie...

This paper investigates the numerical integration error calibration problem in Lie group sigma point filters to obtain more accurate estimation results. On the basis of the theoretical framework of the Bayes–Sard quadrature transformation, we f...

It is shown that, for both compact and non-compact Lie groups, vector-coherent-state methods provide straightforward derivations of holomorphic representations on symmetric spaces. Complementary vector-coherent-state methods are introduced to derive...

In this paper, we investigate the algebraic conditions of algebraic Schouten solitons on three-dimensional Lorentzian Lie groups associated with the perturbed canonical connection and the perturbed Kobayashi–Nomizu connection. Furthermore, we p...

The paper offers a general symbolic method to study the motion in a non-inertial reference frame. In order to achieve this, we use the algebraic and geometric properties of the Lie group of special orthogonal tensors, ${\mathbf{SO}}_3$, and the Lie group of the ri...

This paper aims to test the structure of interest rates during the period from 1 September 1981 to 28 December 2020 by using Lie algebras and groups. The selected period experienced substantial events impacting interest rates, such as the economic cr...

This paper deals with the use of Lie group methods to solve optimization problems in blind signal processing (BSP), including Independent Component Analysis (ICA) and Independent Subspace Analysis (ISA). The paper presents the theoretical fundamental...

In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau’s symplectic model of stati...

We study Lorentzian contact and Lorentzian–Sasakian structures in Hom-Lie algebras. We find that the three-dimensional $\mathfrak{sl}(2,\mathbb{R})$ and Heisenberg Lie algebras provide examples of such structures, respectively. Curvature tensor properties in Lorentz...

We consider a variable coefficient Burgers system in two spatial variables. A similarity mapping is used to reduce the number of independent variables. Lie symmetry analysis is applied to the reduced system. We present the equivalence groups for this...

Let G be a connected Lie group with Lie algebra $\mathbf{g}.$ This review is devoted to studying the fundamental dynamic properties of elements in the normalizer ${\mathfrak{N}}_G$ of G. Through an algebraic characterization of ${\mathfrak{N}}_G$, we analyze the different dynamics inside the...

The Boiti–Leon–Manna–Pempinelli (BLMP) equation with coefficients being functions of time is considered. Since the coefficient functions are arbitrary, we have a class of BLMP equations. Symmetry analysis is carried out for this cla...

This paper addresses the problem of monocular Simultaneous Localization And Mapping on Lie groups using fiducial patterns. For that purpose, we propose a reformulation of the classical camera model as a model on matrix Lie groups. Thus, we define an...

This paper advances a Lie-group approach to measurement by identifying symmetry conditions that determine when effect sizes from different instruments can be meaningfully compared. Measurement transformations are modeled as elements of a two-paramete...

Traditional convolutional neural networks (CNNs) lack equivariance for transformations such as rotation and scaling. Consequently, they typically exhibit weak robustness when an input image undergoes generic transformations. Moreover, the complex mod...

This work establishes a complete algebraic classification of hypersurfaces with totally symmetric cubic form, including the Codazzi, parallel, and totally geodesic cases, on the 4-dimensional 3-step nilpotent Lie group $Ni{l}^4$ endowed with six left-inva...