Hom-Lie group and hom-Lie algebra from Lie group and Lie algebra perspective

2021 ◽  
pp. 2150068
Author(s):  
S. Merati ◽  
M. R. Farhangdoost
Keyword(s):  
Group Theory ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Group Structure ◽  
Point Of View ◽  
Lie Group Theory

A hom-Lie group structure is a smooth group-like multiplication on a manifold, where the structure is twisted by a isomorphism. The notion of hom-Lie group was introduced by Jiang et al. as integration of hom-Lie algebra. In this paper we want to study hom-Lie group and hom-Lie algebra from the Lie group’s point of view. We show that some of important hom-Lie group issues are equal to similar types in Lie groups and then many of these issues can be studied by Lie group theory.

Singularities of mechanisms and the degree of mobility

2003 ◽  
Vol 217 (2) ◽  
pp. 111-119 ◽  
Author(s):  
J Lerbet ◽  
M Fayet
Keyword(s):  
Group Theory ◽  
Lie Algebra ◽  
Lie Group ◽  
Tangent Cone ◽  
Singularity Analysis ◽  
Lie Group Theory ◽  
Singular Configurations ◽  
Statics And Dynamics

The Kinematics, statics and dynamics of singular configurations of mechanisms are analysed. Using Lie group theory, the tangent cone at such a configuration is defined and calculated. It is clear that the structure of the cone is directly linked with one of Lie algebra. The statics and dynamics of singularity analysis are addressed, making it possible to apply the concept of the transitory degree of mobility.

scholarly journals Controllability of affine right-invariant systems on solvable Lie groups

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Yuri L. Sachkov
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Point Of View ◽  
Theoretical Point ◽  
Solvable Lie Groups ◽  
International Audience ◽  
Systems On Lie Groups

International audience The aim of this paper is to present some recent results on controllability of right-invariant systems on Lie groups. From the Lie-theoretical point of view, we study conditions under which subsemigroups generated by half-planes in the Lie algebra of a Lie group coincide with the whole Lie group.

scholarly journals LIE GROUPS AND NUMERICAL SOLUTIONS OF DIFFERENTIAL EQUATIONS: INVARIANT DISCRETIZATION VERSUS DIFFERENTIAL APPROXIMATION

2013 ◽  
Vol 53 (5) ◽  
pp. 438-443 ◽  
Author(s):  
Decio Levi ◽  
Pavel Winternitz
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Group Theory ◽  
Differential Equations ◽  
Numerical Solution ◽  
Lie Groups ◽  
Lie Group ◽  
Numerical Solutions ◽  
Differential Approximation ◽  
Lie Group Theory

We briefly review two different methods of applying Lie group theory in the numerical solution of ordinary differential equations. On specific examples we show how the symmetry preserving discretization provides difference schemes for which the “first differential approximation” is invariant under the same Lie group as the original ordinary differential equation.

Intrinsic Formulation of Dynamics of Curvilinear Systems

2005 ◽  
Author(s):  
Jean Lerbet
Keyword(s):  
Group Theory ◽  
Lie Algebra ◽  
Lie Group ◽  
Mechanical Systems ◽  
Artificial Satellites ◽  
Lie Group Theory ◽  
Classical Models ◽  
To Come ◽  
External Forces ◽  
Scalar Equations

The paper concerns the dynamics of curvilinear systems which are often met in mechanical systems (robots, artificial satellites and so on). We only suppose that each section is rigid. Using Lie group theory, a general curvilinear system is then equivalent to a differentiable distribution of displacements, elements of the Lie group of Euclidean displacements the algebra of which may be identified with the Lie algebra of screws. The kinematics is described by the lagrangian field of deformations and the lagrangian field of velocities elements of the Lie algebra and with standard hypotheses about the distribution of external forces, the intrinsic equations are obtained, the displacements or deformations being small or large. The non linearities (of inertia terms as for internal strenghts) appear by the adjoint mapping and its derivation: the Lie braket. Last, the elements to automatically obtain scalar equations and to come back to more classical models (beam, cable,) are given.

Characteristic functions of semigroups in semi-simple Lie groups

2019 ◽  
Vol 31 (4) ◽  
pp. 815-842
Author(s):  
Luiz A. B. San Martin ◽  
Laercio J. Santos
Keyword(s):  
Laplace Transform ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Convex Cone ◽  
Convex Set ◽  
Flag Manifold ◽  
Riemannian Metric ◽  
Iwasawa Decomposition ◽  
Characteristic Functions

Abstract Let G be a noncompact semi-simple Lie group with Iwasawa decomposition {G=KAN} . For a semigroup {S\subset G} with nonempty interior we find a domain of convergence of the Helgason–Laplace transform {I_{S}(\lambda,u)=\int_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg} , where dg is the Haar measure of G, {u\in K} , {\lambda\in\mathfrak{a}^{\ast}} , {\mathfrak{a}} is the Lie algebra of A and {gu=ke^{\mathsf{a}(g,u)}n\in KAN} . The domain is given in terms of a flag manifold of G written {\mathbb{F}_{\Theta(S)}} called the flag type of S, where {\Theta(S)} is a subset of the simple system of roots. It is proved that {I_{S}(\lambda,u)<\infty} if λ belongs to a convex cone defined from {\Theta(S)} and {u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))} , where {\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}} is a B-convex set and {\pi:K\rightarrow\mathbb{F}_{\Theta(S)}} is the natural projection. We prove differentiability of {I_{S}(\lambda,u)} and apply the results to construct of a Riemannian metric in {\mathcal{D}_{\Theta(S)}(S)} invariant by the group {S\cap S^{-1}} of units of S.

scholarly journals FORMATION CONTROL OF MULTIPLE UNICYCLE-TYPE ROBOTS USING LIE GROUP

2016 ◽  
Vol 56 (1) ◽  
pp. 1
Author(s):  
Youwei Dong ◽  
Ahmed Rahmani
Keyword(s):  
Group Theory ◽  
Numerical Simulations ◽  
Lie Group ◽  
Formation Control ◽  
Directed Acyclic Graph ◽  
Control Law ◽  
Lie Group Theory ◽  
Acyclic Graph ◽  
Communication Topology

In this paper the formation control of a multi-robots system is investigated. The proposed control law, based on Lie group theory, is applied to control the formation of a group of unicycle-type robots. The communication topology is supposed to be a rooted directed acyclic graph and fixed. Some numerical simulations using Matlab are made to validate our results.

scholarly journals Central Extensions of Lie Groups Preserving a Differential Form

2020 ◽  
Author(s):  
Tobias Diez ◽  
Bas Janssens ◽  
Karl-Hermann Neeb ◽  
Cornelia Vizman
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Differential Form ◽  
Integral Form ◽  
Symplectic Manifolds ◽  
Central Extensions ◽  
Mapping Space ◽  
Canonical Class ◽  
Differential Characters

Abstract Let $M$ be a manifold with a closed, integral $(k+1)$-form $\omega $, and let $G$ be a Fréchet–Lie group acting on $(M,\omega )$. As a generalization of the Kostant–Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of ${\mathfrak{g}}$ by ${\mathbb{R}}$, indexed by $H^{k-1}(M,{\mathbb{R}})^*$. We show that the image of $H_{k-1}(M,{\mathbb{Z}})$ in $H^{k-1}(M,{\mathbb{R}})^*$ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of $G$ by the circle group ${\mathbb{T}}$. The idea is to represent a class in $H_{k-1}(M,{\mathbb{Z}})$ by a weighted submanifold $(S,\beta )$, where $\beta $ is a closed, integral form on $S$. We use transgression of differential characters from $ S$ and $ M $ to the mapping space $ C^\infty (S, M) $ and apply the Kostant–Souriau construction on $ C^\infty (S, M) $.

PERTURBATIVE SOLUTIONS OF DIFFERENTIAL EQUATIONS IN LIE GROUPS

2005 ◽  
Vol 02 (01) ◽  
pp. 111-125 ◽  
Author(s):  
PAOLO ANIELLO
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Similarity Transformation ◽  
Perturbative Expansion ◽  
Analytic Perturbation ◽  
The Matrix ◽  
Matrix Formula

We show that, given a matrix Lie group [Formula: see text] and its Lie algebra [Formula: see text], a 1-parameter subgroup of [Formula: see text] whose generator is the sum of an unperturbed matrix Â0 and an analytic perturbation Â♢(λ) can be mapped — under suitable conditions — by a similarity transformation depending analytically on the perturbative parameter λ, onto a 1-parameter subgroup of [Formula: see text] generated by a matrix [Formula: see text] belonging to the centralizer of Â0 in [Formula: see text]. Both the similarity transformation and the matrix [Formula: see text] can be determined perturbatively, hence allowing a very convenient perturbative expansion of the original 1-parameter subgroup.

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