On a degenerate series of unitary representations of a Lie algebra of the group O(p, q)

1970 ◽  
Vol 3 (4) ◽  
pp. 337-339 ◽  
Author(s):  
A. V. Nikolov
Keyword(s):  
Lie Algebra ◽  
Unitary Representations

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>

scholarly journals One more method to construct the irreducible unitary representations of nipotent Lie groups

1986 ◽  
Vol 40 (1) ◽  
pp. 89-94
Author(s):  
A. A. Astaneh
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Nilpotent Lie Group ◽  
Irreducible Unitary Representations ◽  
Simply Connected

AbstractIn this paper one more canonical method to construct the irreducible unitary representations of a connected, simply connected nilpotent Lie group is introduced. Although we used Kirillov' analysis to deduce this procedure, the method obtained differs from that of Kirillov's, in that one does not need to consider the codjoint representation of the group in the dual of its Lie algebra (in fact, neither does one need to consider the Lie algebra of the group, provided one knows certain connected subgroups and their characters). The method also differs from that of Mackey's as one only needs to induce characters to obtain all irreducible representations of the group.

scholarly journals UNITARY REPRESENTATIONS FOR THE SCHRÖDINGER–VIRASORO LIE ALGEBRA

2012 ◽  
Vol 12 (01) ◽  
pp. 1250132 ◽  
Author(s):  
XIUFU ZHANG ◽  
SHAOBIN TAN
Keyword(s):  
Lie Algebra ◽  
Virasoro Algebra ◽  
Unitary Representations

In this paper, conjugate-linear anti-involutions and unitary Harish-Chandra modules over the Schrödinger–Virasoro algebra are studied. It is proved that there are only two classes conjugate-linear anti-involutions over the Schrödinger–Virasoro algebra. The main result of this paper is that a unitary Harish-Chandra module over the Schrödinger–Virasoro algebra is simply a unitary Harish-Chandra module over the Virasoro algebra.

scholarly journals GENERALIZATION OF THE GELL–MANN DECONTRACTION FORMULA FOR sl(n,ℝ) AND su(n) ALGEBRAS

2011 ◽  
Vol 08 (02) ◽  
pp. 395-410 ◽  
Author(s):  
IGOR SALOM ◽  
DJORDJE ŠIJAČKI
Keyword(s):  
Closed Form ◽  
Lie Algebra ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Algebraic Expression ◽  
Matrix Elements ◽  
Group Manifold ◽  
The Matrix

The so-called Gell–Mann or decontraction formula is proposed as an algebraic expression inverse to the Inönü–Wigner Lie algebra contraction. It is tailored to express the Lie algebra elements in terms of the corresponding contracted ones. In the case of sl (n,ℝ) and su (n) algebras, contracted w.r.t. so (n) subalgebras, this formula is generally not valid, and applies only in the cases of some algebra representations. A generalization of the Gell–Mann formula for sl (n,ℝ) and su (n) algebras, that is valid for all tensorial, spinorial, (non)unitary representations, is obtained in a group manifold framework of the SO(n) and/or Spin (n) group. The generalized formula is simple, concise and of ample application potentiality. The matrix elements of the [Formula: see text], i.e. SU(n)/SO(n), generators are determined, by making use of the generalized formula, in a closed form for all irreducible representations.

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