Lie Group Structure Research Articles

Topology of Hankel matrices and applications

In this paper we first construct a Lie group structure on n×n Hankel matrices over R+ by Hadamard product and then we find its Lie algebra structure and finally calculate dimension of this manifold over R+. Moreover, we discuss topological properties of this manifold using Frobenious norm. We pointed out the relation between Lie group and Lie algebra structures of these matrices by exponential map. It is also shown that the Hadamard product on Hankel matrices over R+ is not bounded by Frobenious norm. Lastly, we provide some applications of these manifolds.

Second-order-optimal filtering on SE(2)×R2 for the Chaplygin sleigh

In the last decades many filters have been proposed to solve the problem of pose estimation that naturally arises in robotics, both for manipulators and mobile robots. The dynamics of mobile robots can be mathematically described by exploiting the theory of Lie groups embedding holonomic and nonholonomic constraints. In this paper we design a second-order-optimal filter for the so-called Chaplygin sleigh, that is a mechanical system with a nonholonomic constraint. In particular we examine the importance to know the exact dynamic equations, and to exploit the underline Lie group structure of the system. Moreover, we investigate the conditions that ensure the preservation of the nonholonomic constraint by properly choosing the affine connection which guarantees that the orthogonal velocity is equal to zero. In this work the sensing system consists of a GPS-like configuration (Global Positioning System) obtained by using two antennas attached to the planar rigid body and an INS-like unit (Inertial Navigation System) to measure the velocities.

Jet spaces over Carnot groups

Jet spaces over \mathbb{R}^n have been shown to have a canonical structure of stratified Lie groups (also known as Carnot groups). We construct jet spaces over stratified Lie groups adapted to horizontal differentiation and show that these jet spaces are themselves stratified Lie groups. Furthermore, we show that these jet spaces support a prolongation theory for contact maps, and in particular, a Bäcklund type theorem holds. A byproduct of these results is an embedding theorem that shows that every stratified Lie group of step s + 1 can be embedded in a jet space over a stratified Lie group of step s .

Stokes phenomena, Poisson–Lie groups and quantum groups

Let g be a complex semisimple Lie algebra, b±⊂g a pair of opposite Borel subalgebras, and r∈b−⊗b+ the corresponding solution of the classical Yang–Baxter equations. Let G be the simply–connected Poisson–Lie group corresponding to (g,r), H⊂B±⊂G the subgroups with Lie algebras h=b−∩b+ and b±, and G⁎=B+×HB− the Poisson–Lie group dual of G. G–valued Stokes phenomena were used by Boalch [3,4] to give a canonical, analytic linearisation of the Poisson–Lie group structure on G⁎. Ug–valued Stokes phenomena were used by the first author to construct a twist killing the KZ associator, and therefore give a transcendental construction of the Drinfeld–Jimbo quantum group Uħg[23]. In the present paper, we show that the former construction can be obtained as semiclassical limit of the latter. Along the way, we also show that the R–matrix of Uħg is a Stokes matrix for the dynamical KZ equations.

The Lie Group Structure of Genus Two Hyperelliptic ℘ Functions

We consider the generalized dual transformation for hyperelliptic ℘ functions. For the genus two case, by constructing a quadratic invariant form, we find that hyperelliptic ℘ functions have the SO(3,2) ∼= Sp(4,R)/Z2 Lie group structure. .

Nonexistence of Surjective Lie Homomorphism Between Complex Special and General Linear Groups

Lie groups are very important objects in algebra, topology and geometry, as it is naturally endowed with two special structures: the algebraic structure of group and the geometric structure of differential manifold. Hence, it is quite meaningful to better know the structure of Lie groups and the actions, and the transformations between them. Two specific theorems about Lie groups and their homomorphism and Lie homomorphism are proved in this paper. First, the fact that there does not exist a surjective Lie group homomorphism between and is introduced. This is done by first having a trial of constructing Lie group surjective homomorphism but fails. Next, this result is further proved by applying topological method that to construct a covering map between and , followed by computing the fundamental group of . Finally, by applying that is a simple group, the fact that there does not exist a surjective group homomorphism between and is demonstrated, together with the statement that is a simple group.

Restricted Riemannian geometry for positive semidefinite matrices

We introduce the manifold of restrictedn×n positive semidefinite matrices of fixed rank p, denoted S(n,p)⁎. The manifold itself is an open and dense submanifold of S(n,p), the manifold of n×n positive semidefinite matrices of the same rank p, when both are viewed as manifolds in Rn×n. This density is the key fact that makes the consideration of S(n,p)⁎ statistically meaningful. We furnish S(n,p)⁎ with a convenient, and geodesically complete, Riemannian geometry, as well as a Lie group structure, that permits analytical closed forms for endpoint geodesics, parallel transports, Fréchet means, exponential and logarithmic maps. This task is done partly through utilizing a reduced Cholesky decomposition, whose algorithm is also provided. We produce a second algorithm from this framework to estimate principal eigenspaces and demonstrate its superior performance over other existing algorithms.

The Geometry of Navigation Problems

While many works exploiting an existing Lie group structure have been proposed for state estimation, in particular the invariant extended Kalman filter (IEKF), few papers address the construction of a group structure that allows casting a given system into the framework of invariant filtering. In this article, we introduce a large class of systems encompassing most problems involving a navigating vehicle encountered in practice. For those systems we introduce a novel methodology that systematically provides a group structure for the state space, including vectors of the body frame such as biases. We use it to derive observers having properties akin to those of linear observers or filters. The proposed unifying and versatile framework encompasses all systems, where IEKF has proved successful, improves state-of-the art “imperfect” IEKF for inertial navigation with sensor biases, and allows addressing novel examples, like GNSS antenna lever arm estimation.

On the unit component of the Newman–Unti group

In this short note we identify the unit component of the Newman–Unti (NU) group in the fine very strong topology. In previous work, this component has been endowed with an infinite-dimensional Lie group structure, while the full NU-group does not support such a structure.

Almost Kähler structures on complex hyperbolic space

Complex hyperbolic space has the structure of a solvable Lie group. Based on the recent classification of Riemannian left-invariant metrics we describe all left-invariant almost K?hler structures on that group. The only K?hler structure is the standard structure of the complex hyperbolic space and all others are strictly almost K?hler. We calculated the Chern connection and obtained the characterization by its torsion tensor. It is proved that the only SCF-soliton is the K?hler one.

Nonlinear constructive observer design for direct homography estimation

Feature-based homography estimation approaches rely on extensive image processing for feature extraction and matching, and do not adequately account for the information provided by the image. Therefore, developing efficient direct techniques to extract the homography from images is essential. This paper presents a novel nonlinear direct homography observer that exploits the Lie group structure of SL(3) and its action on the space of image maps. Theoretical analysis demonstrates local asymptotic convergence of the observer. The observer design is also extended for partial measurements of velocity under the assumption that the unknown component is constant or slowly time-varying. Finally, simulation results demonstrate the performance of the proposed solutions on real images.

Shape Analysis of Functional Data With Elastic Partial Matching.

Elastic Riemannian metrics have been used successfully for statistical treatments of functional and curve shape data. However, this usage suffers from a significant restriction: the function boundaries are assumed to be fixed and matched. In practice, functional data often comes with unmatched boundaries. It happens, for example, in dynamical systems with variable evolution rates, such as COVID-19 infection rate curves associated with different geographical regions. Here, we develop a Riemannian framework that allows for partial matching, comparing, and clustering of functions with phase variability and uncertain boundaries. We extend past work by (1) Defining a new diffeomorphism group G over the positive reals that is the semidirect product of a time-warping group and a time-scaling group; (2) Introducing a metric that is invariant to the action of G; (3) Imposing a Riemannian Lie group structure on G to allow for an efficient gradient-based optimization for elastic partial matching; and (4) Presenting a modification that, while losing the metric property, allows one to control the amount of boundary disparity in the registration. We illustrate this framework by registering and clustering shapes of COVID-19 rate curves, identifying basic patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods.

Anomaly detection of time series correlations via a novel Lie group structure

We investigate a natural Lie group structure that correlations possess, interpreted as a special case of the quotient manifold structure for correlation matrices. The one‐dimensional setting of the general theory gives rise to additional structure in the form of a natural associative multiplication of correlations making the space of correlations into a Lie group. We explicitly compute left‐invariant vector fields, exponential and logarithmic mappings, and ultimately, a closed‐form distance formula between correlations. The Lie group formalism is then applied to an application in anomalous correlation detection. The advantage of the Lie group method illustrated with time series data of stock prices.

Weyl-Einstein structures on conformal solvmanifolds

A conformal Lie group is a conformal manifold (M, c) such that M has a Lie group structure and c is the conformal structure defined by a left-invariant metric g on M. We study Weyl-Einstein structures on conformal solvable Lie groups and on their compact quotients. In the compact case, we show that every conformal solvmanifold carrying a Weyl-Einstein structure is Einstein. We also show that there are no left-invariant Weyl-Einstein structures on non-abelian nilpotent conformal Lie groups, and classify them on conformal solvable Lie groups in the almost abelian case. Furthermore, we determine the precise list (up to automorphisms) of left-invariant metrics on simply connected solvable Lie groups of dimension 3 carrying left-invariant Weyl-Einstein structures.

Classification of Left Invariant Riemannian Metrics on Complex Hyperbolic Space

It is well known that \({\mathbb {C}}H^n\) has the structure of a solvable Lie group with left invariant metric of constant holomorphic sectional curvature. In this paper we give the full classification of all possible left invariant Riemannian metrics on this Lie group. We prove that each of those metrics is of constant negative scalar curvature, only one of them being Einstein (up to isometry and scaling).

The half-period addition formulae for genus two hyperelliptic ℘ functions and the Sp(4, ) Lie group structure

In the previous study, by using the two-flows Kowalevski top, we have demonstrated that the genus two hyperelliptic functions provide the Sp(4, )/ ≅ SO(3, 2) Lie algebra structure. In this study, by directly using the differential equations of the genus two hyperelliptic ℘ functions instead of using integrable models, we demonstrate that the half-period addition formula for the genus two hyperelliptic functions provides the order two Sp(4, ) Lie group structure.

Lie theory for asymptotic symmetries in general relativity: The NU group

We study the Newman–Unti (NU) group from the viewpoint of infinite-dimensional geometry. The NU group is a topological group in a natural coarse topology, but it does not become a manifold and hence a Lie group in this topology. To obtain a manifold structure we consider a finer Whitney-type topology. This turns the unit component of the NU group into an infinite-dimensional Lie group. We then study the Lie theoretic properties of this group. Surprisingly, the group operations of the full NU group become discontinuous, whence the NU group does not support a Lie group structure. The NU group contains the Bondi–Metzner–Sachs (BMS) group as a subgroup, whose Lie group structure was constructed in a previous article. It is well known that the NU Lie algebra splits into a direct sum of Lie ideals of the Lie algebras of the BMS group and conformal rescalings of scri. However, the lack of a Lie group structure on the NU group implies that the BMS group cannot be embedded as a Lie subgroup therein.

The 3d O(4) model as an effective approach to the QCD phase diagram

The QCD phase diagram is one of the most prominent outstanding puzzles within the Standard Model. Various experiments, which aim at its exploration beyond small baryon density, are operating or in preparation. From the theoretical side, this is an issue of non-perturbative QCD, and therefore of lattice simulations. However, a finite baryon density entails a technical problem (known as the “sign problem”), which has not been overcome so far. Here we present a study of an effective theory, the O(4) non-linear sigma model. It performs spontaneous symmetry breaking with the same Lie group structure as 2-flavor QCD in the chiral limit, which strongly suggests that they belong to the same universality class. Since we are interested in high temperature, we further assume dimensional reduction to the 3d O(4) model, which implies topological sectors. As pointed out by Skyrme, Wilczek and others, its topological charge takes the role of the baryon number. Hence the baryon chemical potential µB appears as an imaginary vacuum angle, which can be included in the lattice simulation without any sign problem. We present numerical results for the critical line in the chiral limit, and for the crossover in the presence of light quark masses. Their shapes are compatible with other predictions, but up to the value of about µB ≈ 300 MeV we do not find the notorious Critical Endpoint (CEP).

On-manifold probabilistic Iterative Closest Point: Application to underwater karst exploration

This paper proposes MpIC, an on-manifold derivation of the probabilistic Iterative Correspondence (pIC) algorithm, which is a stochastic version of the original Iterative Closest Point. It is developed in the context of autonomous underwater karst exploration based on acoustic sonars. First, a derivation of pIC based on the Lie group structure of is developed. The closed-form expression of the covariance modeling the estimated rigid transformation is also provided. In a second part, its application to 3D scan matching between acoustic sonar measurements is proposed. It is a prolongation of previous work on elevation angle estimation from wide-beam acoustic sonar. While the pIC approach proposed is intended to be a key component in a Simultaneous Localization and Mapping framework, this paper focuses on assessing its viability on a unitary basis. As ground truth data in karst aquifer are difficult to obtain, quantitative experiments are carried out on a simulated karst environment and show improvement compared to previous state-of-the-art approach. The algorithm is also evaluated on a real underwater cave dataset demonstrating its practical applicability.

Nonholonomic Reorientation of Free-Flying Space Robots Using Parallelogram Actuation in Joint Space

A robot with large-degree-of-freedom joints is a promising future space robot capable of adapting to various environments and can perform dexterous tasks with multiple manipulators. The attitude dynamics of a free-flying robot shows nonholonomy, which enables the robot to reorient its attitude by moving its body. However, previous studies were not generally able to handle the nonholonomy, especially for robots with large degrees of freedom of joints. In the present study, we analytically investigate a maneuver in which joints are actuated along a parallelogram trajectory in joint angle space and propose an analytical path modification method in joint angle space in order to achieve the target attitude. Parallelogram actuation guarantees that the body configuration is returned to the initial state after one set of maneuvers, and thus the attitude of the robot is independently reoriented to the target maintaining the body configuration. The analytical solution is provided by applying Magnus expansion to the kinematics equation of rotational matrices, and its Lie group structure contributes to concise mathematical expressions. In addition, the analytical solution does not cause numerical difficulties, such as combinatorial explosion, and so can fully make use of the reorientation ability of a robot with large degrees of freedom.