BARUT–GIRARDELLO COHERENT STATES CONSTRUCTED BY Ym m(θ,φ) AND Ym+1 m(θ,φ) FOR A FREE PARTICLE ON THE SPHERE

2004 ◽  
Vol 19 (35) ◽  
pp. 2619-2628 ◽  
Author(s):  
A. CHENAGHLOU ◽  
H. FAKHRI
Keyword(s):  
Lie Algebra ◽  
Coherent States ◽  
Free Particle ◽  
Irreducible Representations ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Shape Invariance ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Lowering Operator

Using the realization idea of simultaneous shape invariance with respect to two different parameters of the associated Legendre functions, the Hilbert space of spherical harmonics Yn m(θ,φ) corresponding to the motion of a free particle on a sphere is split into a direct sum of infinite-dimensional Hilbert subspaces. It is shown that these Hilbert subspaces constitute irreducible representations for the Lie algebra u (1,1). Then by applying the lowering operator of the Lie algebra u (1,1), Barut–Girardello coherent states are constructed for the Hilbert subspaces consisting of Ym m(θ,φ) and Ym+1 m(θ,φ).

REPRESENTATION OF THE HEISENBERG ALGEBRA h4 BY THE LOWEST LANDAU LEVELS AND THEIR COHERENT STATES

2005 ◽  
Vol 20 (30) ◽  
pp. 2295-2303
Author(s):  
H. FAKHRI ◽  
Z. SHADMAN
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Coherent States ◽  
Rotational Symmetry ◽  
Dynamical Symmetry ◽  
Heisenberg Algebra ◽  
Landau Levels ◽  
Shape Invariance ◽  
Half Axis

Using simultaneous shape invariance with respect to two different parameters, we introduce a pair of appropriate operators which realize shape invariance symmetry for the monomials on a half-axis. It leads to the derivation of rotational symmetry and dynamical symmetry group H4 with infinite-fold degeneracy for the lowest Landau levels. This allows us to represent the Heisenberg–Lie algebra h4 not only by the lowest Landau levels, but also by their corresponding standard coherent states.

FINE DISINTEGRATION OF THE LEFT REGULAR REPRESENTATION

2005 ◽  
Vol 04 (06) ◽  
pp. 683-706 ◽  
Author(s):  
JEAN LUDWIG ◽  
CARINE MOLITOR-BRAUN
Keyword(s):  
Regular Representation ◽  
Unitary Irreducible Representation ◽  
Irreducible Representations ◽  
Left Translation ◽  
Direct Integral ◽  
Translation Invariant ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Left Regular Representation ◽  
Left Regular

Let Hn be the (2n + 1)-dimensional Heisenberg group. We decompose L2(Hn) as the closure of a direct sum of infinitely many left translation invariant eigenspaces (for certain systems of partial differential equations). The restriction of the left regular representation to each one of these eigenspaces disintegrates into a direct integral of unitary irreducible representations, such that each infinite dimensional unitary irreducible representation appears with multiplicity 0 or 1 in this disintegration.

Standard Representations of Simple Lie Algebras

1968 ◽  
Vol 20 ◽  
pp. 344-361 ◽  
Author(s):  
I. Z. Bouwer
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Partial Solution ◽  
Irreducible Representations ◽  
Complex Numbers ◽  
Dominant Weight ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Algebraic Representations

Let L be any simple finite-dimensional Lie algebra (defined over the field K of complex numbers). Cartan's theory of weights is used to define sets of (algebraic) representations of L that can be characterized in terms of left ideals of the universal enveloping algebra of L. These representations, called standard, generalize irreducible representations that possess a dominant weight. The newly obtained representations are all infinite-dimensional. Their study is initiated here by obtaining a partial solution to the problem of characterizing them by means of sequences of elements in K.

scholarly journals A filtration on the ring of Laurent polynomials and representations of the general linear Lie algebra

2021 ◽  
Vol 32 (1) ◽  
pp. 9-32
Author(s):  
C. Choi ◽  
◽  
S. Kim ◽  
H. Seo ◽  
◽  
...  
Keyword(s):  
Lie Algebra ◽  
Irreducible Representations ◽  
General Linear ◽  
Laurent Polynomials ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Weight Multiplicity ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Multiplicity Free

We first present a filtration on the ring Ln of Laurent polynomials such that the direct sum decomposition of its associated graded ring grLn agrees with the direct sum decomposition of grLn, as a module over the complex general linear Lie algebra gl(n), into its simple submodules. Next, generalizing the simple modules occurring in the associated graded ring grLn, we give some explicit constructions of weight multiplicity-free irreducible representations of gl(n).

scholarly journals On Some Infinite Dimensional Representations of Semi-Simple Lie Algebras

1965 ◽  
Vol 25 ◽  
pp. 211-220 ◽  
Author(s):  
Hiroshi Kimura
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Infinite Dimensional ◽  
Complete Reducibility ◽  
Finite Dimensional

Let g be a semi-simple Lie algebra over an algebraically closed field K of characteristic 0. For finite dimensional representations of g, the following important results are known; 1) H1(g, V) = 0 for any finite dimensional g space V. This is equivalent to the complete reducibility of all the finite dimensional representations,2) Determination of all irreducible representations in connection with their highest weights.3) Weyl’s formula for the character of irreducible representations [9].4) Kostant’s formula for the multiplicity of weights of irreducible representations [6],5) The law of the decomposition of the tensor product of two irreducible representations [1].

scholarly journals On the Monodromy of Meromorphic Cyclic Opers on the Riemann Sphere

2020 ◽  
Author(s):  
Charles LeBarron Alley
Keyword(s):  
Lie Algebra ◽  
Riemann Sphere ◽  
Irreducible Representations ◽  
System Of Equations ◽  
Isomonodromic Deformations ◽  
Direct Sum ◽  
Structure Constants ◽  
Monodromy Map

Abstract We study the monodromy of meromorphic cyclic $\textrm{SL}(n,{\mathbb{C}})$-opers on the Riemann sphere with a single pole. We prove that the monodromy map, sending such an oper to its Stokes data, is an immersion in the case where the order of the pole is a multiple of $n$. To do this, we develop a method based on the work of Jimbo, Miwa, and Ueno from the theory of isomonodromic deformations. Specifically, we introduce a system of equations that is equivalent to the isomonodromy equations of Jimbo–Miwa–Ueno, but which is adapted to the decomposition of the Lie algebra $\mathfrak{sl}(n,\mathbb{C})$ as a direct sum of irreducible representations of $\mathfrak{sl}(2,\mathbb{C})$. Using properties of some structure constants for $\mathfrak{sl}(n,\mathbb{C})$ to analyze this system of equations, we show that deformations of certain families of cyclic $\textrm{SL}(n,\mathbb{C})$-opers on the Riemann sphere with a single pole are never infinitesimally isomonodromic.

scholarly journals SU(1,1) solution for the Dunkl–Coulomb problem in two dimensions and its coherent states

2018 ◽  
Vol 33 (20) ◽  
pp. 1850112 ◽  
Author(s):  
M. Salazar-Ramírez ◽  
D. Ojeda-Guillén ◽  
R. D. Mota ◽  
V. D. Granados
Keyword(s):  
Energy Spectrum ◽  
Closed Form ◽  
Lie Algebra ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Two Dimensions ◽  
Irreducible Representations ◽  
Radial Part ◽  
Coulomb Problem ◽  
Radial Schrödinger Equation

We study the radial part of the Dunkl–Coulomb problem in two dimensions and show that this problem possesses the SU(1,[Formula: see text]1) symmetry. We introduce two different realizations of the su(1,[Formula: see text]1) Lie algebra and use the theory of irreducible representations to obtain the energy spectrum and eigenfunctions. For the first algebra realization, we apply the Schrödinger factorization to the radial part of the Dunkl–Coulomb problem to construct the algebra generators. In the second realization, we introduce three operators, one of them proportional to the Hamiltonian of the radial Schrödinger equation. Finally, we use the SU(1,[Formula: see text]1) Sturmian basis to construct the radial coherent states in a closed form.

scholarly journals GENERALIZED COHERENT STATES FOR THE SPHERICAL HARMONICS $Y_{m}^{m}(\theta,\phi)$

2010 ◽  
Vol 25 (10) ◽  
pp. 2165-2170 ◽  
Author(s):  
H. FAKHRI ◽  
B. MOJAVERI
Keyword(s):  
Spherical Harmonics ◽  
Generating Functions ◽  
Coherent States ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Rodrigues Formula ◽  
Generalized Coherent States ◽  
Infinite Sequences

The associated Legendre functions [Formula: see text] for a given l-m, may be taken into account as the increasing infinite sequences with respect to both indices l and m. This allows us to construct the exponential generating functions for them in two different methods by using Rodrigues formula. As an application then we present a scheme to construct generalized coherent states corresponding to the spherical harmonics [Formula: see text].

Dynamical and structural symmetries for the highest Landau levels on the AdS 2

Open Physics ◽  
2012 ◽  
Vol 10 (2) ◽  
Author(s):  
Pakhshan Espoukeh ◽  
Hossein Fakhri
Keyword(s):  
Magnetic Field ◽  
Differential Operators ◽  
Dynamical Symmetry ◽  
Quantization Condition ◽  
Landau Levels ◽  
Irreducible Representations ◽  
Legendre Functions ◽  
Ladder Operators ◽  
Associated Legendre Functions ◽  
Dirac Quantization

AbstractThe aim of the paper is to use the recurrence relations with respect to both indices of the associated Legendre functions for the extraction of the Dirac quantization condition and dynamical symmetry group U(1, 1) corresponding to the highest Landau levels on the hyperbolic plane with uniform magnetic field B. Irreducible representations of the su(2) algebra are obtained by the ladder differential operators which change B by 1/2 unit and mode number by one unit. Two different classes of the irreducible representations of SU(1, 1) with the even and odd boson numbers 2B − 1/2 are extracted for the Bargmann indices 1/4 and 3/4, respectively. Finally, we show that shape invariance symmetry is realized by the ladder operators which shift only the magnetic field B by 1/2 unit.

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