scholarly journals Remarks on the Representation Theory of the Moyal Plane

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
J. M. Isidro ◽  
P. Fernández de Córdoba ◽  
J. M. Rivera-Rebolledo ◽  
J. L. G. Santander
Keyword(s):  
Representation Theory ◽  
Unitary Representation ◽  
Explicit Construction ◽  
Commutator Algebra ◽  
Moyal Plane

We present an explicit construction of a unitary representation of the commutator algebra satisfied by position and momentum operators on the Moyal plane.

scholarly journals NONRELATIVISTIC HOLOGRAPHY — A GROUP-THEORETICAL PERSPECTIVE

2014 ◽  
Vol 29 (03n04) ◽  
pp. 1430001 ◽  
Author(s):  
V. K. DOBREV
Keyword(s):  
Representation Theory ◽  
Theoretical Perspective ◽  
Explicit Construction ◽  
Intertwining Operators ◽  
Difference Operators ◽  
Semisimple Lie Groups ◽  
Boundary Operators ◽  
Boundary Fields ◽  
Theoretical Results ◽  
Schrödinger Algebra

We give a review of some group-theoretical results related to nonrelativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We first recall the interpretation of nonrelativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. One important result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory, and that these operators and the bulk-to-boundary operators are intertwining operators. Further, we recall the fact that there is a hierarchy of equations on the boundary, invariant with respect to Schrödinger algebra. We also review the explicit construction of an analogous hierarchy of invariant equations in the bulk, and that the two hierarchies are equivalent via the bulk-to-boundary intertwining operators. The derivation of these hierarchies uses a mechanism introduced first for semisimple Lie groups and adapted to the nonsemisimple Schrödinger algebra. These require development of the representation theory of the Schrödinger algebra which is reviewed in some detail. We also recall the q-deformation of the Schrödinger algebra. Finally, the realization of the Schrödinger algebra via difference operators is reviewed.

THE CLASSICAL LIMIT OF REPRESENTATION THEORY OF THE QUANTUM PLANE

2013 ◽  
Vol 24 (04) ◽  
pp. 1350031 ◽  
Author(s):  
IVAN C. H. IP
Keyword(s):  
Fourier Transform ◽  
Representation Theory ◽  
Unitary Representation ◽  
Classical Limit ◽  
Mellin Transform ◽  
Tensor Products ◽  
Quantum Plane ◽  
The Fourier Transform

We showed that there is a complete analogue of a representation of the quantum plane [Formula: see text] where |q| = 1, with the classical ax+b group. We showed that the Fourier transform of the representation of [Formula: see text] on [Formula: see text] has a limit (in the dual corepresentation) toward the Mellin transform of the unitary representation of the ax+b group, and furthermore the intertwiners of the tensor products representation has a limit toward the intertwiners of the Mellin transform of the classical ax+b representation. We also wrote explicitly the multiplicative unitary defining the quantum ax+b semigroup and showed that it defines the corepresentation that is dual to the representation of [Formula: see text] above, and also correspond precisely to the classical family of unitary representation of the ax+b group.

scholarly journals Simple Finite-Dimensional Modules and Monomial Bases from the Gelfand-Testlin Patterns

2021 ◽  
pp. 60-65
Author(s):  
Amadou Keita
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Simple Module ◽  
Explicit Construction ◽  
Research Area ◽  
The Past ◽  
Finite Dimensional ◽  
Dimensional Module

One of the most important classes of Lie algebras is sl_n, which are the n×n matrices with trace 0. The representation theory for sl_n has been an interesting research area for the past hundred years and in it, the simple finite-dimensional modules have become very important. They were classified and Gelfand and Tsetlin actually gave an explicit construction of a basis for every simple finite-dimensional module. This paper extends their work by providing theorems and proofs and constructs monomial bases of the simple module.

scholarly journals COMMENTS ON HAMILTONIAN FORMALISM OF AdS/CFT CORRESPONDENCE

1999 ◽  
Vol 14 (02) ◽  
pp. 147-159 ◽  
Author(s):  
TOSHIO NAKATSU ◽  
NAOTO YOKOI
Keyword(s):  
Representation Theory ◽  
Unitary Representation ◽  
Canonical Quantization ◽  
Hamiltonian Formulation ◽  
Hamiltonian Formalism ◽  
Higher Dimensions ◽  
Irreducible Representations ◽  
Boundary Theory ◽  
Boundary Values ◽  
Simple Mapping

As a toy model to search for Hamiltonian formalism of the AdS/CFT correspondence, we examine a Hamiltonian formulation of the AdS2/CFT1 correspondence emphasizing unitary representation theory of the symmetry. In the course of a canonical quantization of the bulk scalars, a particular isomorphism between the unitary irreducible representations in the bulk and boundary theories is found. This isomorphism defines the correspondence of field operators. It states that field operators of the bulk theory are field operators of the boundary theory by taking their boundary values in a specific way. The Euclidean continuation provides an operator formulation on the hyperbolic coordinates system. The associated Fock vacuum of the bulk theory is located at the boundary, thereby identified with the boundary CFT vacuum. The correspondence is interpreted as a simple mapping of the field operators acting on this unique vacuum. Generalization to higher dimensions is speculated.

scholarly journals The unitary representation theory of GL(n) of an infinite discrete field

1989 ◽  
Vol 67 (1) ◽  
pp. 67-81 ◽  
Author(s):  
Roger E. Howe ◽  
Jonathan Rosenberg
Keyword(s):  
Representation Theory ◽  
Unitary Representation
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