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 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 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 Harmonic analysis on the quotient spaces of Heisenberg groups

1991 ◽  
Vol 123 ◽  
pp. 103-117 ◽  
Author(s):  
Jae-Hyun Yang
Keyword(s):  
Quantum Mechanics ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Unitary Representation ◽  
Lie Group ◽  
Theta Series ◽  
Foundations Of Quantum Mechanics ◽  
Nilpotent Lie Group ◽  
Heisenberg Groups ◽  
Quotient Spaces

A certain nilpotent Lie group plays an important role in the study of the foundations of quantum mechanics ([Wey]) and of the theory of theta series (see [C], [I] and [Wei]). This work shows how theta series are applied to decompose the natural unitary representation of a Heisenberg group.

On Asymptotic Behavior of Induced Representations

1982 ◽  
Vol 34 (1) ◽  
pp. 220-232 ◽  
Author(s):  
Larry Baggett ◽  
Keith F. Taylor
Keyword(s):  
Asymptotic Behavior ◽  
Unitary Representation ◽  
Lie Group ◽  
Closed Subgroup ◽  
List Type ◽  
Connected Component ◽  
Induced Representation ◽  
Induced Representations ◽  
Diagonal Subgroup ◽  
Connected Lie Group

This paper is devoted to the proof of the following theorem.THEOREM 1.1. Let H be a closed subgroup of a connected Lie group G, let N denote the largest (closed) subgroup of H which is normal in all of G, and suppose that π is a unitary representation of H whose restriction to N is a multiple of a character χ of N. Then every matrix coefficient of the induced representation Uπ vanishes at infinity modulo the kernel of Uπ providing that the following two conditions hold:i) N is almost-connected (finite modulo its connected component).ii) The subgroup Hk is “regularly related” to the diagonal subgroup D in Gk for at least one integer k ≧ k0 where k0 is determined by G and H.

scholarly journals On Square-Integrable Representations of A Lie Group of 4-Dimensional Standard Filiform Lie Algebra

CAUCHY ◽  
2020 ◽  
Vol 6 (2) ◽  
pp. 84
Author(s):  
Edi Kurniadi
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Unitary Representations ◽  
Identity Operator ◽  
Orbit Method ◽  
Irreducible Unitary Representations ◽  
Dimension 2 ◽  
Filiform Lie Group ◽  
Filiform Lie Algebra

<p class="Abstract">In this paper, we study irreducible unitary representations of a real standard filiform Lie group with dimension equals 4 with respect to its basis. To find this representations we apply the orbit method introduced by Kirillov. The corresponding orbit of this representation is genereric orbits of dimension 2. Furthermore, we show that obtained representation of this group is square-integrable. Moreover, in such case , we shall consider its Duflo-Moore operator as multiple of scalar  identity operator. In our case  that scalar is equal to one.</p>

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 OPERATOR KERNELS FOR IRREDUCIBLE REPRESENTATIONS OF EXPONENTIAL LIE GROUPS

2008 ◽  
Vol 78 (2) ◽  
pp. 301-316
Author(s):  
DETLEV POGUNTKE
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Unitary Representation ◽  
Kernel Function ◽  
Lie Group ◽  
Linear Form ◽  
Irreducible Representations ◽  
Compactly Supported ◽  
Exponential Lie Group ◽  
Exponential Lie Groups

AbstractA nine-dimensional exponential Lie group G and a linear form ℓ on the Lie algebra of G are presented such that for all Pukanszky polarizations 𝔭 at ℓ the canonically associated unitary representation ρ=ρ(ℓ,𝔭) of G has the property that ρ(ℒ1(G)) does not contain any nonzero operator given by a compactly supported kernel function. This example shows that one of Leptin’s results is wrong, and it cannot be repaired.

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 ^ .

On the Unitarized Adjoint Representation of a Semisimple Lie Group II

1977 ◽  
Vol 29 (6) ◽  
pp. 1217-1222
Author(s):  
Ronald L. Lipsman
Keyword(s):  
Lebesgue Measure ◽  
Lie Algebra ◽  
Unitary Representation ◽  
Lie Group ◽  
Adjoint Action ◽  
Adjoint Representation ◽  
Semisimple Lie Group ◽  
Group Ii ◽  
Image Position

Let G be a connected semisimple Lie group with Lie algebra . Lebesgue measure on is invariant under the adjoint action of G; and so there is a natural unitary representation TG of G on L2 given by

scholarly journals Duflo-Moore Operator for The Square-Integrable Representation of 2-Dimensional Affine Lie Group

2020 ◽  
Vol 6 (2) ◽  
pp. 114-122
Author(s):  
Edi Kurniadi ◽  
Nurul Gusriani ◽  
Betty Subartini
Keyword(s):  
Fourier Transform ◽  
Unitary Representation ◽  
Lie Group ◽  
Regular Representation ◽  
Irreducible Unitary Representation ◽  
Integrable Representation ◽  
Square Integrable Representation

In this paper, we study the quasi-regular and the irreducible unitary representation of affine Lie group  of dimension two. First, we prove a sharpening of Fuhr’s work of Fourier transform of quasi-regular representation of . The second, in such the representation of affine Lie group   is square-integrable then we compute its Duflo-Moore operator instead of using Fourier transform as in F hr’s work.

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