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The overall theme of the conference was geometric group theory, interpreted quite broadly. In general, geometric group theory seeks to understand algebraic properties of groups by studying their actions on spaces with various topological and geometric properties; in particular these spaces must have enough structure-preserving symmetry to admit interesting group actions. Although traditionally geometric group theorists have focused on finitely generated (and even finitely presented) countable discrete groups, the techniques that have been developed are now applied to more general groups, such as Lie groups and Kac-Moody groups, and although metric properties of the spaces have played a key role in geometric group theory, other structure such as complex or projective structures and measure-theoretic structures are being used more and more frequently. Mathematics Subject Classification (2010): 20Fxx, 57Mxx. Introduction by the Organisers In addition to discussing the most recent developments within geometric group theory, the meeting also highlighted several dramatic contributions of geometric group theory to other fields. A particular emphasis within the field was studying several classes of groups which exhibit properties of classical examples such as arithmetic groups but are not themselves arithmetic. The idea that a group can be thought of as a geometric object with non-positive or negative curvature is one of the most fundamental ideas in geometric group theory. Curvature conditions have helped us to understand both the general, randomly defined group and specific families of groups arising from topological of 1630 Oberwolfach Report 28/2013 differential-geometric considerations. The focus has recently shifted to variants on these curvature conditions, both those which were defined long ago but not intensively studied and newly introduced notions. For example, Gromov introduced “relative hyperbolicity” at the same time as he defined hyperbolicity, but this was not studied deeply until at least a decade later. Relative hyperbolicity captures behavior similar to that of non-uniform lattices in real hyperbolic spaces in a more general, non-smooth framework. Other variants of hyperbolicity focus on properties of a particular group action rather on the group itself, and generalize classical small cancellation theory. This has led to the construction of quotient groups with prescribed properties, starting from a suitable action of a group on a space, and has had applications to groups arising from unexpected quarters, such as proving that the Cremona group is not simple. Several talks during the week dealt with new techniques and questions. For example, in some talks the use of an auxiliary space with a group action is less central, such as in investigating the possible growth rates of finitely generated groups, or in attempts to establish a general theory of totally disconnected locally compact groups. In others, the structures on spaces preserved by the group action are more of an analytic nature than a geometric one, for example there are some exciting connections with measure theory and with operator algebras, some of which lead to deep topological questions. Specific families of groups that were considered in the talks included mapping class groups MCG(Σ) of surfaces, groups of outer automorphisms Out(Fn) of nonabelian free groups and isometry groups of buildings. These are of particular interest because of their connections with many other areas of mathematics, and because each in its way generalizes the classical examples of linear groups acting on symmetric spaces. The construction of suitable substitutes for the symmetric spaces and the investigation of even the most basic properties are often very difficult. Certainly one of the most exciting developments in the field was the recent use of geometric group theory to solve the last open conjecture on W. Thurston’s famous list of problems on the structure of 3-manifolds. Two speakers gave talks explaining both the geometric group theory and its application to 3-manifolds during the official schedule, and informal sessions were held in the evenings for those who wanted to hear more details. Progress is currently being made on simplifying some of the proofs, and there are many further potential applications of the technology to geometric group theory. We had 52 participants from a wide range of countries, and 26 official lectures. The staff in Oberwolfach was—as always—extremely supportive and helpful. We are very grateful for the additional funding for 5 young PhD students and recent postdocs through Oberwolfach-Leibniz-Fellowships. In addition, there was one young student funded through the DMV Student’s Conference. We think that this provided a great opportunity for these students. We feel that the meeting was exciting and highly successful. The quality of all lectures was outstanding, and outside of lectures there was a constant buzz Geometric Structures in Group Theory 1631 of intense mathematical conversations. We are confident that this conference will lead to both new and exciting mathematical results and to new collaborations. Geometric Structures in Group Theory 1633 Workshop: Geometric Structures in Group Theory

Elastic Riemannian metrics have been used successfully in the past for statistical treatments of functional and curve shape data. However, this usage has suffered from an important restriction: the function boundaries are assumed fixed and matched. Functional data exhibiting unmatched boundaries typically arise from dynamical systems with variable evolution rates such as COVID-19 infection rate curves associated with different geographical regions. In this case, it is more natural to model such data with sliding boundaries and use partial matching, i.e., only a part of a function is matched to another function. Here, we develop a comprehensive Riemannian framework that allows for partial matching, comparing, and clustering of functions under both phase variability and uncertain boundaries. We extend past work by: (1) Forming a joint action of the time-warping and time-scaling groups; (2) Introducing a metric that is invariant to this joint action, allowing for a gradient-based approach to elastic partial matching; and (3) Presenting a modification that, while losing the metric property, allows one to control relative influence of the two groups. This framework is illustrated for registering and clustering shapes of COVID-19 rate curves, identifying essential patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods., Submitted to IEEE Transactions on Pattern Analysis and Machine Intelligence

This paper systematically investigates the exact solutions to an extended (2+1)-dimensional Boussinesq equation, which arises in several physical applications, including the propagation of shallow-water waves, with the help of the Lie symmetry analysis method. We acquired the vector fields, commutation relations, optimal systems, two stages of reductions, and exact solutions to the given equation by taking advantage of the Lie group method. The method plays a crucial role to reduce the number of independent variables by one in each stage and finally forms an ODE which is solved by taking relevant suppositions and choosing the arbitrary constants that appear therein. Furthermore, Lie symmetry analysis (LSA) is implemented for perceiving the symmetries of the Boussinesq equation and then culminating the solitary wave solutions. The behavior of the obtained results for multiple cases of symmetries is obtained in the present framework and demonstrated through three-and two-dimensional dynamical wave profiles. These solutions show single soliton, multiple solitons, elastic behavior of combo soliton profiles, and stationary waves, as can be seen from the graphics. The outcomes of the present investigation manifest that the considered scheme is systematic and significant to solve nonlinear evolution equations.

For achieving the desired configuration of spacecraft at the desired fixed time, a sub-optimal fixed-finite-horizon configuration control method on the Lie group SE(3) is developed based on the Model Predictive Static Programming (MPSP). The MPSP technique has been widely used to solve finite-horizon optimal control problems and is known for its high computational efficiency thanks to the closed-form solution, but it cannot be directly applied to systems on SE(3). The methodological innovation in this paper enables that the MPSP technique is extended to the geometric control on SE(3), using the variational principle, the left-invariant properties of Lie groups, and the topology structure of Lie algebra space. Moreover, the energy consumption, which is crucial for spacecraft operations, is considered as the objective function to be optimized in the optimal control formulation. The effectiveness of the designed sub-optimal control method is demonstrated through an online simulation under disturbances and state measurement errors.

A flat quadratic Lie algebra ( g , 〈 , 〉 , k ) is a Lie algebra g endowed with a flat pseudo-Euclidean metric 〈 , 〉 and a quadratic structure k. In geometrical terms, it is a Lie algebra of a Lie group endowed with both a flat left-invariant pseudo-Riemannian metric and a bi-invariant metric. It is known that if a Lie algebra admits a symplectic form and a quadratic structure then it is a flat quadratic Lie algebra. In this paper, we show that there exist non-abelian Lie algebras which admit a flat Lorentzian metric and a quadratic structure. If g is such Lie algebra, then we show that g is a trivial central extension of R ⋊ H 2 k + 1 and [ g , g ] ≃ H 2 k + 1 , where H 2 k + 1 is the Heisenberg Lie algebra and k ≥ 2 . In particular, we prove that g is 3-step solvable, not nilpotent and hence admits no symplectic structure. We show that there exists only one non-abelian Lie algebra of dimension n ≤ 6 which admits a flat Lorentzian metric and a quadratic structure. Using a variant of the double extension method, which conserves both the flat and the quadratic metrics, we prove that any Lie algebra which admits a flat Lorentzian metric and a quadratic structure, is a double extension of an abelian Lie algebra and we construct a large class of such (non-abelian) Lie algebras in higher dimensions. Oscillator Lie algebras denoted by g λ constitute an important class of quadratic Lie algebras which are of the form R ⋊ H 2 k + 1 . It is known that g λ admits no flat Lorentzian metric. We show in this paper that g λ admits a Ricci-flat Lorentzian metric. The paper also study the class of quadratic 2-step nilpotent Lie algebras ( g , k ) which constitute examples of flat quadratic Lie algebras since k is also flat in this case.

Mccarthy, P. J., & Nielsen, C. (2021). Global Sampled-Data Regulation of a Class of Fully Actuated Invariant Systems on Simply Connected Nilpotent Matrix Lie Groups. IEEE Transactions on Automatic Control, 1–1. https://doi.org/10.1109/TAC.2021.3058053

The existing convolutional neural network (CNN) models have shown excellent performance in remote sensing scene classification. However, the structure of such models is becoming more and more complex, and the learning of low-level features is difficult to interpret. To address this problem, in this study, we introduce lie group machine learning into the CNN model, try to combine both approaches to extract more distinguishing ability and effective features, and propose a novel network model, namely, the lie group regional influence network (LGRIN). First, manifold space samples of the lie group are obtained by mapping, and then, the features of the lie group are extracted after the operations of image decomposition and integral image calculation. Second, the multidilation pooling is integrated into the CNN architecture. At the same time, the image regional influence network module is designed to guide the attention of the classification model by using the regional-level supervision of the decomposition. Finally, the fusion features are classified, and the predicted results are obtained. Our model takes full advantage of regional influence, lie group kernel function, and lie group feature learning. Moreover, our model produces satisfactory performance on three public and challenging data sets: Aerial Image Dataset (AID), UC Merced, and NWPU-RESISC45. The experimental results verify that, compared with the state-of-the-art methods, this method is more explanatory and achieves higher accuracy.

The current work deals with the classical and non-classical Lie group analysis for the fractional type system of nonlinear Chen–Lee–Liu (CLL) equations. The invariance surface condition is imposed to this system of fractional differential equations to find non-classical generators. Non-classical and classical Lie symmetry analysis are used to similarity reductions of this system. Corresponding exact solutions are obtained for the extracted generators in both classical and non-classical senses. Convergence of the obtained solutions when fractional order tends to the integer one, is graphically checked. Finally, the conservation laws are investigated for the system of time-fractional CLL equations, specially for the non-classical vector fields. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.

We present Chebyshev type cubature rules for the exact integration of rational symmetric functions with poles on prescribed coordinate hyperplanes. Here the integration is with respect to the densities of unitary Jacobi ensembles stemming from the Haar measures of the orthogonal and the compact symplectic Lie groups., Comment: 10 pages, 1 table

After three decades of computational multibody system (MBS) dynamics, current research is centered at the development of compact and user-friendly yet computationally efficient formulations for the analysis of complex MBS. The key to this is a holistic geometric approach to the kinematics modeling observing that the general motion of rigid bodies and the relative motion due to technical joints are screw motions. Moreover, screw theory provides the geometric setting and Lie group theory the analytic foundation for an intuitive and compact MBS modeling. The inherent frame invariance of this modeling approach gives rise to very efficient recursive $O ( n ) $ algorithms, for which the so-called “spatial operator algebra” is one example, and allows for use of readily available geometric data. In this paper, three variants for describing the configuration of tree-topology MBS in terms of relative coordinates, that is, joint variables, are presented: the standard formulation using body-fixed joint frames, a formulation without joint frames, and a formulation without either joint or body-fixed reference frames. This allows for describing the MBS kinematics without introducing joint reference frames and therewith rendering the use of restrictive modeling convention, such as Denavit–Hartenberg parameters, redundant. Four different definitions of twists are recalled, and the corresponding recursive expressions are derived. The corresponding Jacobians and their factorization are derived. The aim of this paper is to motivate the use of Lie group modeling and to provide a review of different formulations for the kinematics of tree-topology MBS in terms of relative (joint) coordinates from the unifying perspective of screw and Lie group theory.

This paper examines the use of Lie group and Lie Algebra theory to construct the geometry of pairwise comparisons matrices. The Hadamard product (also known as coordinatewise, coordinate-wise, elementwise, or element-wise product) is analyzed in the context of inconsistency and inaccuracy by the decomposition method. The two designed components are the approximation and orthogonal components. The decomposition constitutes the theoretical foundation for the multiplicative pairwise comparisons.

We focus on the role of symmetries in geometric control theory from the Hamiltonian viewpoint. We demonstrate the power of Ian Anderson's package Differential Geometry in CAS software Maple for dealing with control problems on Lie groups. We apply the tools to the problem of vertical rolling disc, however, anyone can modify our approach and tools to other control problems. (C) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to the normalizer property or to uniqueness of Sylow subgroups. As a consequence, we show algebraic decompositions of o-minimal nilpotent groups, and we prove that a nilpotent Lie group is definable in an o-minimal expansion of the reals if and only if it is a linear algebraic group., 12 pages

In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space a satisfying three coupled classical dynamical Yang–Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on a , generalising asymptotic boundary Knizhnik–Zamolodchikov–Bernard (KZB) equations. The recipe involves folding and contracting a -invariant and θ -twisted symmetric classical dynamical r -matrices along an involutive automorphism θ . In case of the universal enveloping algebra of a simple Lie algebra g we determine the subclass of Schiffmann’s classical dynamical r -matrices which are a -invariant and θ -twisted. The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories.

Let S be a subsemigroup of a simply connected nilpotent Lie group G. We construct an asymptotic semigroup S0 in the associated graded Lie group G0 of G. We can compute the image of S0 in the abelianization $$ {G}_0^{\mathrm{ab}}={G}^{\mathrm{ab}}. $$ G 0 ab = G ab . This gives useful information about S. As an application, we obtain a transparent proof of the following result of E. B. Vinberg and the author: either there is an epimorphism f : G → ℝ such that f (s) ≥ 0 for every s in S or the closure $$ \overline{S} $$ S ¯ of S is a subgroup of G and $$ G/\overline{S} $$ G / S ¯ is compact.

We investigate the interaction between a singular surface and a strong shock in the self-gravitating interstellar gas clouds with the assumption of spherical symmetry. Using the method of the Lie group of transformations, a particular solution of the flow variables and the cooling–heating function for an infinitely strong shock is obtained. This paper explores an application of the singular surface theory in the evolution of an acceleration wave front propagating through an unperturbed medium. We discuss the formation of an acceleration, considering the cases of compression and expansion waves. The influence of the cooling–heating function on a shock formation is explained. The results of a collision between a strong shock and an acceleration wave are discussed using the Lax evolutionary conditions.

This article presents a brief historical review of G. I. Taylor's solution of the point blast wave problem which was applied to the Trinity test of the first atomic bomb. Lie group symmetry techniques are used to derive Taylor's famous two-fifths law that relates the position of a blast wave to the time after the explosion and the total energy released. The theory of exterior differential systems is combined with the method of characteristics to demonstrate that the solution of the blast wave problem is directly related to the basic relationships that exist between the geometry (or symmetry) and the physics of wave propagation through the equations of motion. The point blast wave model is cast in terms of two exterior differential systems and both systems are shown to be integrable with local solutions for the velocity, pressure, and density along curves in space and time behind the blast wave.

In this paper, we study $$\eta$$ η -Ricci solitons on almost cosymplectic $$(k,\mu )$$ ( k , μ ) -manifolds. As an application, it is proved that if an almost cosymplectic $$(k,\mu )$$ ( k , μ ) -metric with $$k k < 0 represents a Ricci soliton, then the potential vector field of the Ricci soliton is a strict infinitesimal contact transformation, and the corresponding almost cosymplectic manifold is locally isometric to a Lie group whose local structure is determined completely by $$k k < 0 . In addition, a concrete example is constructed to illustrate the above result.

Let $(Z,\omega)$ be a connected Kahler manifold with an holomorphic action of the complex reductive Lie group $U^{\mathbb C}$, where $U$ is a compact connected Lie group acting in a hamiltonian fashion. Let $G$ be a closed compatible Lie group of $U^{\mathbb C}$ and let $M$ be a $G$-invariant connected submanifold of $Z$. Let $x\in M$. If $G$ is a real form of $U^{\mathbb C}$, we investigate conditions such that $G\cdot x$ compact implies $U^{\mathbb C} \cdot x$ is compact as well. The vice-versa is also investigated. We also characterize $G$-invariant real submanifolds such that the norm square of the gradient map is constant. As an application, we prove a splitting result for real connected submanifolds of $(Z,\omega)$ generalizing a result proved in \cite{pg}, see also \cite{bg,bs}., Comment: 9 opages

Let $N$ be a connected and simply connected nilpotent Lie group, $K$ be a compact subgroup of $Aut(N)$, the group of automorphisms of $N$ and $\delta$ be a class of unitary irreducible representations of $K$. The triple $(N,K,\delta)$ is a commutative triple if the convolution algebra $\mathfrak{U}_{\delta}^{1}(N)$ of $\delta$-radial integrable functions is commutative. In this paper, we obtain first a parametrization of $\delta$ spherical functions by means of the unitary dual $\widehat{N}$ and then an inversion formula for the spherical transform of $F\in \mathfrak{U}_{\delta}^{1}(N)$.

Solitary waves are usually outcome of nonlinear and dispersive effects. The Date–Jimbo–Kashiwara–Miwa equation is dispersive and highly nonlinear soliton equation. The Date–Jimbo– Kashiwara–Miwa equation is a well-known integrable generalization of the KP equation and represents evolution of propagating waves in nonlinear dispersive media. It has wave dispersion changing properties in nonlinear dynamics. In this work, we aim to carry out symmetry reductions and exact solutions of the Date–Jimbo–Kashiwara–Miwa equation using Lie point symmetries. The possible infinitesimal generators and infinite-dimensional algebra of symmetries are constructed by utilizing the invariance property of Lie groups. Then, similarity variables are used to reduce the test problem and lead to determining ordinary differential equations. These equations provide desired exact solutions under some parametric constraints. The derived solutions are generalized than previous established results and have never been reported yet. Moreover, the self-adjoint equation and conserved vectors with associated symmetries are established under the Lagrangian formulation. To demonstrate the significance of interaction phenomena, the solutions have been extended with numerical simulation. Consequently, stripe soliton, line soliton, multisoliton, soliton fission, parabolic profile and their elastic nature are discussed systematically to validate these solutions with physical phenomena.

We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the Hörmander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp Hörmander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.

We develop a discrete-time optimal control framework for systems evolving on Lie groups. Our article generalizes the original differential dynamic programming method, by employing a coordinate-free, Lie-theoretic approach for its derivation. A key element lies, specifically, in the use of quadratic expansion schemes for cost functions and dynamics defined on Lie groups. The obtained algorithm iteratively optimizes local approximations of the control problem, until reaching a (sub)optimal solution. On the theoretical side, we also study the conditions under which convergence is attained. Details about the behavior and implementation of our method are provided through a simulated example on $TSO(3)$ .

The key problem of remote sensing image classification is to understand the semantic content of the images effectively. Up to now, most methods are to improve the accuracy of the final classification through the convolutional neural networks model. However, the structure of the whole model is complex and contains a large number of parameters. To address this problem, in this study, we proposed a lightweight method of the intrinsic mean within Lie groups, which can achieve high accuracy. We apply the computational advantage of Lie group manifold space and design the kernel function that is suitable for both Lie group sample and vector sample. Experiments are performed on two publicly available and challenging data sets [aerial image dataset (AID) and Northwestern Polytechnical University-REmote Sensing Image Scene Classification, contains 45 scene classes (NWPU-RESISC45)], and the results show that our method is more accurate than the state-of-the-art methods.

A pseudo-Riemannian Lie group ( G , 〈 ⋅ , ⋅ 〉 ) is a connected and simply connected Lie group with a left-invariant pseudo-Riemannian metric of type ( p , q ) . This paper is to study pseudo-Riemannian Lie groups with non-Killing conformal vector fields induced by derivations which is an extension from non-Killing left-invariant conformal vector fields. First we prove that a Riemannian (i.e. type ( n , 0 ) ), Lorentzian (i.e. type ( n − 1 , 1 ) ) or trans-Lorentzian (i.e. type ( n − 2 , 2 ) ) Lie group with such a vector field is solvable. Then we construct non-solvable unimodular pseudo-Riemannian Lie groups with such vector fields for any min ⁡ ( p , q ) ≥ 3 . Finally, we give the classification for the Riemannian and Lorentzian cases.

In this paper, we consider coupled Korteweg-de Vries equations that model the propagation of shallow water waves, ion-acoustic waves in plasmas, solitons, and nonlinear perturbations along internal surfaces between layers of different densities in stratified fluids, for example propagation of solitons of long internal waves in oceans. The method of Lie group analysis is used to on the system to obtain symmetry reductions. Soliton solutions are constructed by use of a linear combination of time and space translation symmetries. Furthermore, we compute conservation laws in two ways that is by multiplier method and by an application of new conservation theorem developed by Nail Ibragimov.

We consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components, that (1) the topological Hopf algebra of holomorphic functions on $G$ is holomorphically reflexive if and only if $G$ is linear; (2) the dual cocommutative topological Hopf algebra of exponential analytic functional on $G$ is holomorphically reflexive. We give a counterexample, which shows that the first criterion cannot be extended to the case of infinitely many components. Nevertheless, we conjecture that, in general, the question can be solved in terms of the Banach-algebra linearity of $G$., version 6, Appendix is added, \circe is replaced by \bullet, plus some minor correction

A group G is called bounded if every conjugation-invariant norm on G has finite diameter. We introduce various strengthenings of this property and investigate them in several classes of groups including semisimple Lie groups, arithmetic groups and linear algebraic groups. We provide applications to Hamiltonian dynamics., Comment: 28 pages; revised version; accepted in Journal of Topology and Analysis

By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of $\mathbb{R}$ or $\mathbb{C}$. We prove an adelic version of superrigidity which implies that two such commensurability classes define the same profinite commensurability class if and only if the algebraic groups are adelically isomorphic. We discuss noteworthy consequences on profinite rigidity questions., 21 pages, revised version

Many bundle gerbes constructed in practice are either infinite-dimensional, or finite-dimensional but built using submersions that are far from being fibre bundles. Murray and Stevenson proved that gerbes on simply-connected manifolds, built from finite-dimensional fibre bundles with connected fibres, always have a torsion $DD$-class. In this note I prove an analogous result for a wide class of gerbes built from principal bundles, relaxing the requirements on the fundamental group of the base and the connected components of the fibre, allowing both to be nontrivial. This has consequences for possible models for basic gerbes, the classification of crossed modules of finite-dimensional Lie groups, the coefficient Lie-2-algebras for higher gauge theory on principal 2-bundles, and finite-dimensional twists of topological $K$-theory., Comment: 18+1 pages. v3 Extended main result much further, additional nontrivial examples. v2 strengthened main result with a simpler proof

Static action recognition in images is challenging, because the image lacks the motion information to characterize the relations between the human and objects. Existing works detect the human with related objects or transfer the motion from videos to images. However the interaction is implicitly depicted. In this paper, we try to solve this problem from a different aspect of view, i.e., to explicitly learn the interactive information from the pose of the human and objects. Humans have different poses in different actions, and the objects in different actions can have different spatial interactions with certain parts of the human. This interaction in poses can be represented by Lie-group naturally. The Lie-group method computes the orientation and distance between the key points or joints, which reveal the relation between humans and objects in different actions. In the experiment, the proposed method shows competitive classification results on several still action image datasets, which advocates the way to recognize still actions by using poses.

In this paper, we introduce left-invariant (strict) nearly para-Kӓhler structures on Lie groups (nearly para- Kӓhler Lie algebras) and present some properties of them. We prove the existence of a unique connection in terms of the characteristic connection on these Lie groups with totally skew-symmetric torsion tensor and we show that the Nijenhuis tensor is parallel with respect to the characteristic connection. We determine conditions that allow the curvature tensor of the characteristic connection satisfies the first Bianchi identity. Finally, we focus on anti-abelian almost para-complex structures on Lie groups and we study nearly para- Kӓhler conditions for these structures. In this study we encounter bi-invariant metrics as pseudo-Riemannian metrics that accept nearly para- Kӓhler structures. Moreover, we present some examples of nearly para- Kӓhler Lie algebras.

Let $(X,T^{1,0}X)$ be a $(2n+1+d)$-dimensional compact CR manifold with codimension $d+1$, $d\geq1$, and let $G$ be a $d$-dimensional compact Lie group with CR action on $X$ and $T$ be a globally defined vector field on $X$ such that $\mathbb C TX=T^{1,0}X\oplus T^{0,1}X\oplus\mathbb C T\oplus\mathbb C\underline{\mathfrak{g}}$, where $\underline{\mathfrak{g}}$ is the space of vector fields on $X$ induced by the Lie algebra of $G$. In this work, we show that if $X$ is strongly pseudoconvex in the direction of $T$ and $n\geq 2$, then there exists a $G$-equivariant CR embedding of $X$ into $\mathbb C^N$, for some $N\in\mathbb N$. We also establish a CR orbifold version of Boutet de Monvel's embedding theorem., Comment: 36 pages

In response to the lack of specific demonstration and analysis of the research on the necessity of the Lie group strapdown inertial integrated navigation error model based on the Euler angle, two common integrated navigation systems, strapdown inertial navigation system/global navigation satellite system (SINS/GNSS) and strapdown inertial navigation system/doppler velocity log (SINS/DVL), are used as subjects, and the piecewise constant system (PWCS) matrix, based on the Lie group error model, is established. From three aspects of variance estimation, the observability and performance of the system with large misalignment angles for low, medium, and high accuracy levels, traditional error model, Lie group left error model, and right error model are compared. The necessity of research on Lie group error model is analyzed quantitatively and qualitatively. The experimental results show that Lie group error model has better stability of variance estimation, estimation accuracy, and observability than traditional error model, as well as higher practical value.

This paper describes the quasi-static formulation of frictionless line contact between flexible beams by employing the mortar finite element approach. Contact constraints are enforced in a weak sense along the contact region using Lagrange multipliers. A simple projection appropriate for thin beams with circular cross-sections is proposed for the computation of contact regions. It is combined with the geometrically exact beam formalism on the Lie group $SE(3)$ S E ( 3 ) . Interestingly, this framework leads to a constraint gradient and a tangent stiffness invariant under rigid body transformations. The formulation is tested in some numerical examples.

Let $TM$ be a tangent bundle over a Riemannian manifold $M$ with a Riemannian metric $g$ and $TG$ be a tangent Lie group over a Lie group with a left-invariant metric $g$. The purpose of the paper is two folds. Firstly, we study statistical structures on the tangent bundle $TM$ equipped with two Riemannian $g$-natural metrics and lift connections. Secondly, we define a left-invariant complete lift connection on the tangent Lie group $TG$ equipped with metric $\tilde{g}$ introduced in [F. Asgari and H. R. Salimi Moghaddam, On the Riemannian geometry of tangent Lie groups, Rend. Circ. Mat. Palermo II. Series, 2018] and study statistical structures in this setting.

We study left-invariant conformal Killing $2$- or $3$-forms on simply connected $2$-step nilpotent Riemannian Lie groups. We show that if the center of the group is of dimension greater than or equal to 4, then every such form is automatically coclosed (i.e. it is a Killing form). In addition, we prove that the only Riemannian 2-step nilpotent Lie groups with center of dimension at most 3 and admitting left-invariant non-coclosed conformal Killing $2$- and $3$-forms are: the Heisenberg Lie groups and their trivial 1-dimensional extensions, endowed with any left-invariant metric, and the simply connected Lie group corresponding to the free 2-step nilpotent Lie algebra on 3 generators, with a particular 1-parameter family of metrics. The explicit description of the space of conformal Killing $2$- and $3$-forms is provided in each case., 20 pages

In this paper, we study the problem of the existence of left invariant Ricci flat metrics on nilpotent Lie groups. We mainly prove that any nilpotent Lie algebra obtained by a double extension of an Abelian Lie algebra admits at least one left invariant Ricci flat metric. As an application, we obtain certain new nilpotent Lie algebras which admit left invariant Ricci flat metrics.

We construct and study solitonic representations of the conformal net associated to some vacuum Positive Energy Representation (PER) of a loop group LG. For the corresponding solitonic states, we prove the Quantum Null Energy Condition (QNEC) and the Bekenstein Bound. As an intermediate result, we show that a Positive Energy Representation of a loop group LG can be extended to a PER of $$H^{s}(S^1,G)$$ H s ( S 1 , G ) for $$s>3/2$$ s > 3 / 2 , where G is any compact, simple and simply connected Lie group. We also show the existence of the exponential map of the semidirect product $$LG \rtimes R$$ L G ⋊ R , with R a one-parameter subgroup of $$\mathrm{Diff}_+(S^1)$$ Diff + ( S 1 ) , and we compute the adjoint action of $$H^{s+1}(S^1,G)$$ H s + 1 ( S 1 , G ) on the stress energy tensor.

In this paper, we study deformations of complex structures on Lie algebras and its associated deformations of Dolbeault cohomology classes. A complete deformation of complex structures is constructed in a way similar to the Kuranishi family. The extension isomorphism is shown to be valid in this case. As an application, we prove that given a family of left invariant deformations $\{M_t\}_{t\in B}$ of a compact complex manifold $M=(\Gamma\setminus G, J)$ where $G$ is a Lie group, $\Gamma$ a sublattice and $J$ a left invariant complex structure, the set of all $t\in B$ such that the Dolbeault cohomology on $M_t$ may be computed by left invariant tensor fields is an analytic open subset of $B$.

For a semisimple real Lie group $G$ with an irreducible representation $\rho$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $\rho$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense in $\rho(G)$ and that is free, nonabelian and acts properly discontinuously on $V$. This new criterion is more general than the one given in the author's previous paper "Proper affine actions in non-swinging representations" (submitted; available at arXiv:1605.03833), insofar as it also deals with "swinging" representations. We conjecture that it is actually a necessary and sufficient criterion, applicable to all representations., Comment: This paper generalizes the author's previous papers arXiv:1406.5906 and arXiv:1605.03833 . The structure of the proof is similar; a few passages are borrowed from the earlier papers. This version differs from the previous one only by a reference that I added

In this paper, Lie group representations on Hilbert spaces are studied in relation with operator ranges. Let R \mathcal {R} be an operator range of a Hilbert space H \mathcal {H} . Given the set Λ \Lambda of R \mathcal {R} -invariant operators, and given a Lie group representation ρ : G → GL ( H ) \rho :G\rightarrow \text {GL}(\mathcal {H}) , we discuss the induced semigroup homomorphism ρ ~ : ρ − 1 ( Λ ) → B ( R ) \widetilde {\rho }: \rho ^{-1}(\Lambda ) \rightarrow \mathcal {B(R)} for the operator range topology on R \mathcal {R} . In one direction, we work under the assumption ρ − 1 ( Λ ) = G \rho ^{-1} (\Lambda ) = G , so ρ ~ : G → B ( R ) \widetilde {\rho }:G\rightarrow \mathcal {B}(\mathcal {R}) is in fact a group representation. In this setting, we prove that ρ ~ \widetilde {\rho } is continuous (and smooth) if and only if the tangent map d ρ d\rho is R \mathcal {R} -invariant. In another direction, we prove that for the tautological representations of unitary or invertible operators of an arbitrary infinite-dimensional Hilbert space H \mathcal {H} , the set ρ − 1 ( Λ ) \rho ^{-1}(\Lambda ) is neither a group for a large set of nonclosed operator ranges R \mathcal {R} nor closed for all nonclosed operator ranges R \mathcal {R} . Both results are proved by means of explicit counterexamples.

Derivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces $$(M=G/H,g)$$ whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (G/H, g), such that G is one of the compact classical Lie groups $${{\,\mathrm{SO}\,}}(n)$$ , $$\mathrm{U}(n)$$ , and H is a diagonally embedded product $$H_1\times \cdots \times H_s$$ , where $$H_j$$ is of the same type as G. This class includes spheres, Stiefel manifolds, Grassmann manifolds and real flag manifolds. The present work is a contribution to the study of g.o. spaces (G/H, g) with H semisimple.

Recently, two of the authors of this paper constructed cyclic cocycles on Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect all information in the $K$-theory of the corresponding group $C^*$-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions., 40 pages; updates based on referee comments; expanded proof of Proposition 3.3

Domain adaptation aims to minimize the mismatch between the source domain in which models are trained and the target domain to which those models are applied. Most existing works focus on instance reweighting, feature representation, and classifier learning independently, which are ineffective when the domain discrepancy is substantially large. In this study, we propose a new unified hybrid approach that takes advantage of Lie group theory, weighted distribution alignment, and manifold alignment, which are referred to as Lie Group Manifold Analysis (LGMA). LGMA mainly finds a one-parameter sub-group decided by the Lie algebra elements of the intrinsic mean of all samples, and this one-parameter sub-group is a geodesic on the original Lie group. Moreover, the Lie group samples are projected onto the geodesics to maximize the separability of the projected samples for realizing discrimination in the nonlinear Lie group manifold space. As far as we know, LGMA is the first attempt to perform Lie algebra transformation to project the original features in the Lie group space onto Lie algebra manifold space for domain adaptation. Comprehensive experiments validate that our approach considerably outperforms competitive methods on real-world datasets.

Every simply connected and connected solvable Lie group $G$ admits a simply transitive action on a nilpotent Lie group $H$ via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups $G$ can act simply transitive on which Lie groups $H$. So far the focus was mainly on the case where $G$ is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action $\rho: G \to \operatorname{Aff}(H)$ is simply transitive by looking only at the induced morphism $\varphi: \mathfrak{g} \to \operatorname{aff}(\mathfrak{h})$ between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group $G$ acts simply transitive on a given nilpotent Lie group $H$, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension $4$., Comment: 22 pages, 8 tables. Comments are welcome

In this paper, we show that for a connected compact Lie group to be acceptable, it is necessary and sufficient that its derived subgroup is isomorphic to a direct product of the groups $${\text {SU}}(n)$$ , $${\text {Sp}}(n)$$ , $${\text {SO}}(2n+1)$$ , $${\text {G}}_2$$ , $${\text {SO}}(4)$$ . We show that there are invariant functions on $${\text {SO}}_{4}({\mathbb {C}})^{2}$$ which are not generated by 1-argument invariants, though the group $${\text {SO}}_{4}({\mathbb {C}})$$ is acceptable.