Discrete series of unitary representations of the Lie algebra of the group O(p, q)

1968 ◽  
Vol 2 (1) ◽  
pp. 94-95 ◽  
Author(s):  
A. V. Nikolov
Keyword(s):  
Lie Algebra ◽  
Discrete Series ◽  
Unitary Representations

scholarly journals ON THE GEOMETRY OF SIEGEL–JACOBI DOMAINS

2011 ◽  
Vol 08 (08) ◽  
pp. 1783-1798 ◽  
Author(s):  
S. BERCEANU ◽  
A. GHEORGHE
Keyword(s):  
Discrete Series ◽  
Unitary Representations ◽  
Fock Spaces ◽  
Holomorphic Discrete Series ◽  
Orthonormal Bases ◽  
Jacobi Group ◽  
Representation Spaces

We study the holomorphic unitary representations of the Jacobi group based on Siegel–Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel–Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel–Jacobi disk is constructed and polynomial orthonormal bases of the representation spaces are given.

IRREDUCIBLE UNITARY REPRESENTATIONS OF SUp,q II: THE CONTINUOUS SERIES

2001 ◽  
Vol 12 (01) ◽  
pp. 37-47 ◽  
Author(s):  
RAJ WILSON ◽  
ELIZABETH TANNER
Keyword(s):  
Kronecker Product ◽  
Discrete Series ◽  
Unitary Representations ◽  
Continuous Series ◽  
Maximal Dimension ◽  
Cartan Subgroup ◽  
Integrable Functions ◽  
Irreducible Unitary Representations ◽  
Square Integrable Functions ◽  
Vector Part

A class of irreducible unitary representations belonging to the continuous series of SUp,q is explicitly determined in a space composed of the Kronecker product of three spaces of square integrable functions. The continuous series corresponds to a Cartan subgroup whose vector part has maximal dimension. These representations are distinguished by a parameter r = 1, 2, …, p for p ≤ q in SUp,q. For r = 0, one obtains the representations in the discrete series as in [5], and all representations in the continuous series, for r ≠ 0, are obtained explicitly.

scholarly journals Landau levels for the (2 + 1) Dunkl–Klein–Gordon oscillator

2021 ◽  
pp. 2150066
Author(s):  
R. D. Mota ◽  
D. Ojeda-Guillén ◽  
M. Salazar-Ramírez ◽  
V. D. Granados
Keyword(s):  
Magnetic Field ◽  
Energy Spectrum ◽  
External Magnetic Field ◽  
Lie Algebra ◽  
Landau Levels ◽  
Unitary Representations ◽  
Partial Derivatives ◽  
Klein Gordon ◽  
The Magnetic Field

In this paper, we study the (2 + 1)-dimensional Klein–Gordon oscillator coupled to an external magnetic field, in which we change the standard partial derivatives for the Dunkl derivatives. We find the energy spectrum (Landau levels) in an algebraic way, by introducing three operators that close the su(1, 1) Lie algebra and from the theory of unitary representations. Also, we find the energy spectrum and the eigenfunctions analytically, and we show that both solutions are consistent. Finally, we demonstrate that when the magnetic field vanishes or when the parameters of the Dunkl derivatives are set to zero, our results are adequately reduced to those reported in the literature.

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>

IRREDUCIBLE UNITARY REPRESENTATIONS OF SUp,q I: THE DISCRETE SERIES

2001 ◽  
Vol 12 (01) ◽  
pp. 1-36 ◽  
Author(s):  
RAJ WILSON ◽  
ELIZABETH TANNER
Keyword(s):  
Differential Operators ◽  
Discrete Series ◽  
Compact Subgroup ◽  
Holomorphic Functions ◽  
Irreducible Unitary Representation ◽  
Explicit Construction ◽  
Unitary Representations ◽  
Infinitesimal Generators ◽  
Algebraic Properties ◽  
Irreducible Unitary Representations

A class of irreducible unitary representations in the discrete series of SUp,q is explicitly determined in a space of holomorphic functions of three complex matrices. The discrete series, which is the set of all square integrable representations, corresponds to a compact subgroup of SUp,q. The relevant algebraic properties of the group SUp,q are discussed in detail. For a degenerate irreducible unitary representation an explicit construction of the infinitesimal generators of the Lie algebra [Formula: see text] in terms of differential operators is given.

scholarly journals ON THE CHIRAL RINGS IN N=2 AND N=4 SUPERCONFORMAL ALGEBRAS

1993 ◽  
Vol 08 (02) ◽  
pp. 301-324 ◽  
Author(s):  
MURAT GÜNAYDIN
Keyword(s):  
Discrete Series ◽  
Special Property ◽  
Unitary Representations ◽  
Holomorphic Discrete Series ◽  
Triple Systems ◽  
Chiral Rings ◽  
Coset Spaces ◽  
Discrete Series Representations ◽  
Jordan Triple Systems ◽  
Infinite Set

We study the chiral primary rings of N=2 and N=4 superconformal algebras (SCA’s) constructed over triple systems. The chiral primary states of N=2 SCA’s realized over Hermitian Jordan triple systems are given. Their coset spaces G/H are Hermitian-symmetric and can be compact or noncompact. In the noncompact case under the requirement of unitarity of the representations of G, we find an infinite discrete set of chiral primary states associated with the holomorphic discrete series representations of G and their analytic continuation. A further requirement that the corresponding N=2 module be unitary truncates this infinite set to a finite subset. There are no chiral primary states associated with the other unitary representations of noncompact groups. Remarkably, the only noncompact groups G that admit holomorphic discrete series unitary representations are such that their quotients G/H with their maximal compact subgroups H are Hermitian-symmetric. The chiral primary states of N=2 SCA’s constructed over the Freudenthal triple systems are also studied. These algebras have the special property that they admit an extension to N=4 superconformal algebras with the gauge group SU(2) ⊗ SU(2) ⊗ U(1). We then generalize the concept of chiral rings to these maximal N=4 superconformal algebras. We find four different rings associated with each sector (left- or right-moving). Inclusion of both sectors gives 16 different rings. We also show that our analysis yields all the possible rings of N=4 SCA’s.

scholarly journals UNITARY REPRESENTATIONS OF W INFINITY ALGEBRAS

1992 ◽  
Vol 07 (25) ◽  
pp. 6339-6355 ◽  
Author(s):  
SATORU ODAKE
Keyword(s):  
Central Charge ◽  
Discrete Series ◽  
Free Field ◽  
Unitary Representations ◽  
Continuous Series ◽  
Highest Weight ◽  
Free Field Realization ◽  
Highest Weight Representations

We study the irreducible unitary highest weight representations, which are obtained from free field realizations, of W infinity algebras [Formula: see text] with central charges (2, 1, 3, 2M, N, 2M+N). The characters of these representations are computed. We construct a new extended superalgebra [Formula: see text], whose bosonic sector is [Formula: see text]. Its representations obtained from a free field realization with central charge 2M+N, are classified into two classes: continuous series and discrete series. For the former there exists a supersymmetry, but for the latter a supersymmetry exists only for M=N.

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