Geometry Of Lie Groups Research Articles

A “magic” approach to octonionic Rosenfeld spaces

In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfortunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems he had been using do not actually hold true in the case of algebras that are not alternative nor power-associative. A more rigorous approach to the definition of all the planes presented more than thirty years ago by Rosenfeld in terms of their isometry group, can be considered within the theory of coset manifolds, which we exploit in this work, by making use of all real forms of Magic Squares of order three and two over Hurwitz normed division algebras and their split versions. Within our analysis, we find seven pseudo-Riemannian symmetric coset manifolds which seemingly cannot have any interpretation within Rosenfeld’s framework. We carry out a similar analysis for Rosenfeld lines, obtaining that there are a number of pseudo-Riemannian symmetric cosets which do not have any interpretation á la Rosenfeld.

Lie group geometry. Invariant metrics and dynamical systems, dual algebra, and their applications in the group analysis of a one-dimensional kinetic equation

Lie group geometry. Invariant metrics and dynamical systems, dual algebra, and their applications in the group analysis of a one-dimensional kinetic equation

Lie Group Equivariant Convolutional Neural Network Based on Laplace Distribution

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 model structure complicates the interpretation of learned low- and mid-level features. To address these issues, we introduce a Lie group equivariant convolutional neural network predicated on the Laplace distribution. This model’s Lie group characteristics blend multiple mid- and low-level features in image representation, unveiling the Lie group geometry and spatial structure of the Laplace distribution function space. It efficiently computes and resists noise while capturing pertinent information between image regions and features. Additionally, it refines and formulates an equivariant convolutional network appropriate for the Lie group feature map, maximizing the utilization of the equivariant feature at each level and boosting data efficiency. Experimental validation of our methodology using three remote sensing datasets confirms its feasibility and superiority. By ensuring a high accuracy rate, it enhances data utility and interpretability, proving to be an innovative and effective approach.

Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces

Abstract We study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient M ≃ G / K {M\simeq G/K} . Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.

Pseudo‐Riemannian ‐manifolds with dimension at most 21

AbstractLet G2(2) be the non‐compact connected simple Lie group of type G2 over , and let M be a connected analytic complete pseudo‐Riemannian manifold that admits an isometric G2(2)‐action with a dense orbit. For the case , we provide a full description of the manifold M, its geometry and its G2(2)‐action. The latter are always given in terms of a Lie group geometry related to G2(2), and in one case M is essentially the quotient of SO0(3, 4) by a lattice.

On the existence of bi-invariant Finsler metrics on Lie groups

In this paper, we study the geometry of Lie groups with bi-invariant Finsler metrics. We first show that every compact Lie group admits a bi-invariant Finsler metric. Then, we prove that every compact connected Lie group is a symmetric Finsler space with respect to the bi-invariant absolute homogeneous Finsler metric. Finally, we show that if G is a Lie group endowed with a bi-invariant Finsler metric, then, there exists a bi-invariant Riemanninan metric on G such that its Levi-Civita connection coincides the connection of F.

Distinguished orbits of reductive groups

We prove a generalization of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group $G$ acting linearly and rationally on a real vector space $V$. $G$ can be viewed as the real points of a complex reductive group $G^\mathbb C$ which acts on $V^\mathbb C := V \otimes \mathbb C$. Borel-Harish-Chandra show that $G^\mathbb C \cdot v \cap V$ is a finite union of $G$-orbits; moreover, $G^\mathbb C \cdot v$ is closed if and only if $G\cdot v$ is closed. We show that the same result holds not just for closed orbits but for the so-called distinguished orbits. An orbit is called distinguished if it contains a critical point of the norm squared of the moment map on projective space. Our main result compares the complex and real settings to show $G\cdot v$ is distinguished if and only if $G^\mathbb C \cdot v$ is distinguished. In addition, we show that if an orbit is distinguished, then under the negative gradient flow of the norm squared of the moment map the entire $G$-orbit collapses to a single $K$-orbit. This result holds in both the complex and real settings. We finish with some applications to the study of the left-invariant geometry of Lie groups.

Coordinated Motion Design on Lie Groups

The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics. It thereby shows that coordination can be studied in a systematic way once the Lie group geometry of the configuration space is well characterized. Applying the proposed general methodology to particular examples allows to retrieve control laws that have been proposed in the literature on intuitive grounds. A link with Brockett's double bracket flows is also made. The concepts are illustrated on SO(3) , SE(2) and SE(3) .

A note on the geometry of Lie groups

The degenerate semi-Riemannian geometry of a Lie group is studied. Then a naturally reductive homogeneous semi-Riemannian space is obtained from the Lie group in a natural way.

Degenerations of Lie algebras and geometry of Lie groups

Each point of the variety of real Lie algebras is naturally identified with a left invariant Riemannian metric on a Lie group. We study the interplay between invariant-theoretic and Riemannian aspects of this variety. In particular, using the special critical point behavior of certain natural functional on the variety, we determine all the Lie groups which can be endowed with only one left invariant metric up to isometry and scaling, proving first that they correspond to Lie algebras whose only degeneration is to the abelian one. We also find all the Lie algebras which degenerate to the Lie algebra of the hyperbolic space, and all the possible degenerations for 3-dimensional real Lie algebras, by using well known descriptions of left invariant metrics satisfying some pinching curvature conditions. Finally, as another interaction, the closed S L( n)-orbits on the variety are classified, and explicit curves of Einstein solvmanifolds are provided by using curves of closed orbits of the representation Λ 2S L( m)⊗S L( n).

Unsupervised neural learning on lie group.

The present paper aims at introducing the concepts and mathematical details of unsupervised neural learning with orthonormality constrains. The neural structures considered are single non-linear layers and the learnable parameters are organized in matrices, as usual, which gives the parameters spaces the geometrical structure of the Euclidean manifold. The constraint of orthonormality for the connection-matrices further restricts the parameters spaces to differential manifolds such as the orthogonal group, the compact Stiefel manifold and its extensions. For these reasons, the instruments for characterizing and studying the behavior of learning equations for these particular networks are provided by the differential geometry of Lie groups. In particular, two sub-classes of the general Lie-group learning theories are studied in detail, dealing with first-order (gradient-based) and second-order (non-gradient-based) learning. Although the considered class of learning theories is very general, in the present paper special attention is paid to unsupervised learning paradigms.

Generalized Lax pairs, the modified classical Yang-Baxter equation, and affine geometry of Lie groups

We generalize the usual Lax equationd/dt L=[M, L] byd/dt L=−ϱ(M)L, where ϱ is an arbitrary representation of a Lie algebra g (the values ofM) in a representation spaceV (the values ofL). The usual classicalr-matrix programme for Hamiltonian integrable systems is generalized tor-matrices taking values in g⊗V. Ther-matrices are then considered as left invariant torsion-free covariant derivatives on a Lie groupK (with Lie algebraV*). The Classical Yang-Baxter Equation (CYBE) is equivalent to the flatness ofK whereas the Modified CYBE implies thatK is an affine locally symmetric space. An example is discussed.