scholarly journals Ruang Fase Tereduksi Grup Lie Aff (1)

2021 ◽  
Vol 3 (2) ◽  
pp. 180-186
Author(s):  
Edi Kurniadi
Keyword(s):  
Phase Space ◽  
Lie Algebra ◽  
Group Action ◽  
Lie Group ◽  
Dual Space ◽  
Coadjoint Orbits ◽  
Coadjoint Representation

ABSTRAKDalam artikel ini dipelajari ruang fase tereduksi dari suatu grup Lie khususnya untuk grup Lie affine  berdimensi 2. Tujuannya adalah untuk mengidentifikasi ruang fase tereduksi dari  melalui orbit coadjoint buka di titik tertentu pada ruang dual  dari aljabar Lie . Aksi dari grup Lie    pada ruang dual  menggunakan representasi coadjoint. Hasil yang diperoleh adalah ruang Fase tereduksi  tiada lain adalah orbit coadjoint-nya yang buka di ruang dual . Selanjutnya, ditunjukkan pula bahwa grup Lie affine     tepat mempunyai dua buah orbit coadjoint buka.  Hasil yang diperoleh dalam penelitian ini dapat diperluas untuk kasus grup Lie affine  berdimensi  dan untuk kasus grup Lie lainnya.ABSTRACTIn this paper, we study a reduced phase space for a Lie group, particularly for the 2-dimensional affine Lie group which is denoted by Aff (1). The work aims to identify the reduced phase space for Aff (1) by open coadjoint orbits at certain points in the dual space aff(1)* of the Lie algebra aff(1). The group action of Aff(1) on the dual space aff(1)* is considered using coadjoint representation. We obtained that the reduced phase space for the affine Lie group Aff(1) is nothing but its open coadjoint orbits. Furthermore, we show that the affine Lie group Aff (1) exactly has two open coadjoint orbits in aff(1)*. Our result can be generalized for the n(n+1) dimensional affine Lie group Aff(n) and for another Lie group.

scholarly journals Representasi Unitar Tak Tereduksi Grup Lie Dari Aljabar Lie Filiform Real Berdimensi 5

2020 ◽  
Vol 17 (1) ◽  
pp. 100-108
Author(s):  
E Kurniadi
Keyword(s):  
Harmonic Analysis ◽  
Lie Algebra ◽  
Unitary Representation ◽  
Lie Group ◽  
Dual Space ◽  
Irreducible Unitary Representation ◽  
Coadjoint Orbits ◽  
Dimensional Representation ◽  
Induced Representation ◽  
Filiform Lie Algebra

In this paper, we study a harmonic analysis of a Lie group  of a real filiform Lie algebra of dimension 5. Particularly, we study its  irreducible unitary representation (IUR) and contruct this IUR corresponds to its coadjoint orbits through coadjoint actions of its group to its dual space.  Using induced representation of  a 1-dimensional representation of its subgroup we obtain its IUR of its Lie group

scholarly journals A Drift-Free Left Invariant Control System on the Lie Group SO(3)×R3×R3

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Camelia Pop
Keyword(s):  
Control System ◽  
Numerical Integration ◽  
Lie Algebra ◽  
Lie Group ◽  
Poisson Structure ◽  
Dual Space ◽  
Free System ◽  
Geometrical Properties ◽  
Free Left ◽  
Invariant Control

A controllable drift-free system on the Lie group G=SO(3)×R3×R3 is considered. The dynamics and geometrical properties of the corresponding reduced Hamilton’s equations on g∗,·,·- are studied, where ·,·- is the minus Lie-Poisson structure on the dual space g∗ of the Lie algebra g=so(3)×R3×R3 of G. The numerical integration of this system is also discussed.

scholarly journals Relativistic Symmetries and Hamiltonian Formalism

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1810
Author(s):  
Piotr Kosiński ◽  
Paweł Maślanka
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Lorentz Transformation ◽  
Lie Group ◽  
Hamiltonian Formalism ◽  
Coadjoint Orbits ◽  
Natural Consequence ◽  
Canonical Variables ◽  
Conformal Symmetries

The relativistic (Poincaré and conformal) symmetries of classical elementary systems are briefly discussed and reviewed. The main framework is provided by the Hamiltonian formalism for dynamical systems exhibiting symmetry described by a given Lie group. The construction of phase space and canonical variables is given using the tools from the coadjoint orbits method. It is indicated how the “exotic” Lorentz transformation properties for particle coordinates can be derived; they are shown to be the natural consequence of the formalism.

scholarly journals Traces for Star Products on the Dual of a Lie Algebra

2003 ◽  
Vol 15 (05) ◽  
pp. 425-445 ◽  
Author(s):  
Pierre Bieliavsky ◽  
Simone Gutt ◽  
Martin Bordemann ◽  
Stefan Waldmann
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Star Product ◽  
Coadjoint Orbits ◽  
Star Products ◽  
Compact Lie Group ◽  
Regular Orbit ◽  
Elementary Argument

In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the ad-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.

scholarly journals The invariant polynomial algebras for the groups IU(n) and ISO(n)

1984 ◽  
Vol 94 ◽  
pp. 43-59 ◽  
Author(s):  
Hitoshi Kaneta
Keyword(s):  
Lie Algebras ◽  
Lie Group ◽  
Dual Space ◽  
Polynomial Algebra ◽  
Invariant Polynomial ◽  
Coadjoint Representation ◽  
Finitely Generated ◽  
Polynomial Algebras ◽  
Enveloping Algebras ◽  
Connected Lie Group

By the coadjoint representation of a connected Lie group G with the Lie algebra g we mean the representation CoAd(g) = tAd(g-1) in the dual space g*. Imitating Chevalley’s argument for complex semi-simple Lie algebras, we shall show that the CoAd (G)-invariant polynomial algebra on g* is finitely generated by algebraically independent polynomials when G is the inhomogeneous linear group IU(n) or ISO(n). In view of a well-known theorem [8, p. 183] our results imply that the centers of the enveloping algebras for the (or the complexified) Lie algebras of these groups are also finitely generated. Recently much more inhomogeneous groups have been studied in a similar context [2]. Our results, however, are further reaching as far as the groups IU(n) and ISO(n) are concerned [cf. 3, 4, 6, 7, 9].

Fundamental Vector Fields

2020 ◽  
pp. 87-96
Author(s):  
Loring W. Tu
Keyword(s):  
Vector Field ◽  
Fixed Points ◽  
Lie Algebra ◽  
Group Action ◽  
Lie Group ◽  
Vector Fields ◽  
Principal Bundle ◽  
Left Action ◽  
Fundamental Vector ◽  
Vertical Tangent

This chapter addresses fundamental vector fields. The concept of a connection on a principal bundle is essential in the construction of the Cartan model. To define a connection on a principal bundle, one first needs to define the fundamental vector fields. When a Lie group acts smoothly on a manifold, every element of the Lie algebra of the Lie group generates a vector field on the manifold called a fundamental vector field. On a principal bundle, the fundamental vectors are precisely the vertical tangent vectors. In general, there is a relation between zeros of fundamental vector fields and fixed points of the group action. Unless specified otherwise (such as on a principal bundle), a group action is assumed to be a left action.

scholarly journals Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation

2020 ◽  
Author(s):  
Jun Jiang ◽  
◽  
Satyendra Kumar Mishra ◽  
Yunhe Sheng ◽  
◽  
...  
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Group Action ◽  
Lie Group ◽  
Smooth Manifold ◽  
Adjoint Representation ◽  
Lie Group Action

In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this Hexp map. We also describe a Hom-Lie group action on a smooth manifold. Subsequently, we give the notion of an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie algebra (gl(V),[.,.],Ad), and the derivation Hom-Lie algebra of a Hom-Lie algebra.

scholarly journals Another proof for the continuity of the Lipsman mapping

2020 ◽  
Vol 72 (7) ◽  
pp. 945-951
Author(s):  
A. Messaoud ◽  
A. Rahali
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Semidirect Product ◽  
Lie Group ◽  
Inner Product ◽  
Coadjoint Orbits ◽  
Compact Lie Group ◽  
Real Vector ◽  
Finite Dimensional ◽  
Unitary Dual

UDC 515.1 We consider the semidirect product G = K ⋉ V where K is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space V equipped with an inner product 〈 , 〉 . By G ^ we denote the unitary dual of G and by 𝔤 ‡ / G the space of admissible coadjoint orbits, where 𝔤 is the Lie algebra of G . It was pointed out by Lipsman that the correspondence between G ^ and 𝔤 ‡ / G is bijective. Under some assumption on G , we give another proof for the continuity of the orbit mapping (Lipsman mapping) Θ : 𝔤 ‡ / G - → G ^ .

ε‎-Invariance

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Algebra ◽  
Lie Group

The discussions up to Chapter 4 have been concerned with the Lie group. In this chapter, the Lie algebra is constructed by defining the operators ∇ and ∇̃.

scholarly journals On the automorphism group of a symplectic half-flat 6-manifold

2019 ◽  
Vol 31 (1) ◽  
pp. 265-273
Author(s):  
Fabio Podestà ◽  
Alberto Raffero
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Group Action ◽  
Cohomogeneity One ◽  
Cohomogeneity One Action

Abstract We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat {\mathrm{SU}(3)} -structure has Abelian Lie algebra with dimension bounded by {\min\{5,b_{1}(M)\}} . Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on {T\mathbb{S}^{3}} which are invariant under a cohomogeneity one action of {\mathrm{SO}(4)} .

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