Representations of Virasoro-Heisenberg Algebras and Virasoro-Toroidal Algebras

1999 ◽  
Vol 51 (3) ◽  
pp. 523-545 ◽  
Author(s):  
Marc A. Fabbri ◽  
Frank Okoh
Keyword(s):  
Fock Space ◽  
Virasoro Algebra ◽  
Heisenberg Algebra ◽  
Geometric Lattice ◽  
Direct Products ◽  
Complex Vector ◽  
Heisenberg Algebras ◽  
Rank One ◽  
Hyperbolic Lattice ◽  
Completely Reducible

AbstractVirasoro-toroidal algebras, , are semi-direct products of toroidal algebras and the Virasoro algebra. The toroidal algebras are, in turn, multi-loop versions of affine Kac-Moody algebras. Let Γ be an extension of a simply laced lattice by a hyperbolic lattice of rank two. There is a Fock space V(Γ) corresponding to Γ with a decomposition as a complex vector space: V(Γ) = . Fabbri and Moody have shown that when m ≠ 0, K(m) is an irreducible representation of . In this paper we produce a filtration of -submodules of K(0). When L is an arbitrary geometric lattice and n is a positive integer, we construct a Virasoro-Heisenberg algebra . Let Q be an extension of by a degenerate rank one lattice. We determine the components of V(Γ) that are irreducible -modules and we show that the reducible components have a filtration of -submodules with completely reducible quotients. Analogous results are obtained for . These results complement and extend results of Fabbri and Moody.

scholarly journals Heisenberg Algebra in the Bargmann-Fock Space with Natural Cutoffs

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Maryam Roushan ◽  
Kourosh Nozari
Keyword(s):  
Uncertainty Principle ◽  
Fock Space ◽  
Generalized Uncertainty Principle ◽  
Heisenberg Algebra ◽  
Minimal Length ◽  
Maximal Momentum

We construct a Heisenberg algebra in Bargmann-Fock space in the presence of natural cutoffs encoded as minimal length, minimal momentum, and maximal momentum through a generalized uncertainty principle.

scholarly journals A QUADRATIC DEFORMATION OF THE HEISENBERG–WEYL AND QUANTUM OSCILLATOR ENVELOPING ALGEBRAS

1993 ◽  
Vol 08 (20) ◽  
pp. 3479-3493 ◽  
Author(s):  
JENS U. H. PETERSEN
Keyword(s):  
Virasoro Algebra ◽  
Heisenberg Algebra ◽  
Roots Of Unity ◽  
Quantum Oscillator ◽  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Quadratic Deformation ◽  
Two Parameter

A new two-parameter quadratic deformation of the quantum oscillator algebra and its one-parameter deformed Heisenberg subalgebra are considered. An infinite-dimensional Fock module representation is presented, which at roots of unity contains singular vectors and so is reducible to a finite-dimensional representation. The semicyclic, nilpotent and unitary representations are discussed. Witten's deformation of sl 2 and some deformed infinite-dimensional algebras are constructed from the 1d Heisenberg algebra generators. The deformation of the centerless Virasoro algebra at roots of unity is mentioned. Finally the SL q(2) symmetry of the deformed Heisenberg algebra is explicitly constructed.

scholarly journals The 3-point Virasoro algebra and its action on a Fock space

2016 ◽  
Vol 57 (3) ◽  
pp. 031702 ◽  
Author(s):  
Ben Cox ◽  
Elizabeth Jurisich ◽  
Renato A. Martins
Keyword(s):  
Fock Space ◽  
Virasoro Algebra

scholarly journals The N = 1 super Heisenberg–Virasoro vertex algebra at level zero

2021 ◽  
Author(s):  
Dražen Adamović ◽  
Berislav Jandrić ◽  
Gordan Radobolja
Keyword(s):  
Fock Space ◽  
Free Field ◽  
Virasoro Algebra ◽  
Vertex Algebra ◽  
Verma Modules ◽  
Minimal Models ◽  
Highest Weight ◽  
Level Zero ◽  
Free Field Realization ◽  
Highest Weight Representations

We study the representation theory of the [Formula: see text] super Heisenberg–Virasoro vertex algebra at level zero, which extends the previous work on the Heisenberg–Virasoro vertex algebra [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342; D. Adamović and G. Radobolja, Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero, Commun. Contemp. Math. 21(2) (2019) 1850008; Y. Billig, Representations of the twisted Heisenberg–Virasoro algebra at level zero, Can. Math. Bull. 46(4) (2003) 529–537] to the super case. We calculated all characters of irreducible highest weight representations by investigating certain Fock space representations. Quite surprisingly, we found that the maximal submodules of certain Verma modules are generated by subsingular vectors. The formulas for singular and subsingular vectors are obtained using screening operators appearing in a study of certain logarithmic vertex algebras [D. Adamović and A. Milas, On W-algebras associated to [Formula: see text] minimal models and their representations, Int. Math. Res. Notices 2010(20) (2010) 3896–3934].

Representations of Quantum Heisenberg Algebras

1994 ◽  
Vol 46 (5) ◽  
pp. 920-929 ◽  
Author(s):  
Marc A. Fabbri ◽  
Frank Okoh
Keyword(s):  
Heisenberg Algebra ◽  
Roots Of Unity ◽  
Direct Sums ◽  
Von Neumann ◽  
One Dimensional ◽  
Infinite Dimensional ◽  
Root Of Unity ◽  
Heisenberg Algebras ◽  
Neumann Theorem ◽  
Von Neumann Theorem

AbstractA Lie algebra is called a Heisenberg algebra if its centre coincides with its derived algebra and is one-dimensional. When is infinite-dimensional, Kac, Kazhdan, Lepowsky, and Wilson have proved that -modules that satisfy certain conditions are direct sums of a canonical irreducible submodule. This is an algebraic analogue of the Stone-von Neumann theorem. In this paper, we extract quantum Heisenberg algebras, q(), from the quantum affine algebras whose vertex representations were constructed by Frenkel and Jing. We introduce the canonical irreducible q()-module Mq and a class Cq of q()-modules that are shown to have the Stone-von Neumann property. The only restriction we place on the complex number q is that it is not a square root of 1. If q1 and q2 are not roots of unity, or are both primitive m-th roots of unity, we construct an explicit isomorphism between q1() and q2(). If q1 is a primitive m-th root of unity, m odd, q2 a primitive 2m-th or a primitive 4m-th root of unity, we also construct an explicit isomorphism between q1() and q2().

The Vacuum Distributions of the Truncated Virasoro Fields are Products of Gamma Distributions

2017 ◽  
Vol 24 (01) ◽  
pp. 1750004 ◽  
Author(s):  
Luigi Accardi ◽  
Andreas Boukas ◽  
Yun-Gang Lu
Keyword(s):  
Fourier Transform ◽  
Characteristic Function ◽  
Heisenberg Group ◽  
Lie Algebra ◽  
Fock Space ◽  
Random Variables ◽  
Virasoro Algebra ◽  
Commutation Relations ◽  
Gamma Distributions

In a recent paper, using a splitting formula for the multi-dimensional Heisenberg group, we derived a formula for the vacuum characteristic function (Fourier transform) of quantum random variables defined as self-adjoint sums of Fock space operators satisfying the multidimensional Heisenberg Lie algebra commutation relations. In this paper we use that formula to compute the characteristic function of quantum random variables defined as suitably truncated sums of the Virasoro algebra generators. By relating the structure of the Virasoro fields to the quadratic quantization program and using techniques developed in that context we prove that the vacuum distributions of the truncated Virasoro fields are products of independent, but not identically distributed, shifted Gamma-random variables.

Paraquantum strings in noncommutative space–time

2015 ◽  
Vol 30 (30) ◽  
pp. 1550175 ◽  
Author(s):  
M. A. Seridi ◽  
N. Belaloui
Keyword(s):  
Fock Space ◽  
Mass Operator ◽  
Virasoro Algebra ◽  
String Model ◽  
Closed String ◽  
Noncommutative Space ◽  
Photon State ◽  
Open Strings ◽  
Mass Degeneracy ◽  
Match Level

A parabosonic string is assumed to propagate in a total noncommutative target phase space. Three models are investigated: open strings, open strings between two parallel [Formula: see text] branes and closed ones. This leads to a generalization of the oscillators algebra of the string and the corresponding Virasoro algebra. The mass operator is no more diagonal in the ordinary Fock space, a redefinition of this later will modify the mass spectrum, so that, neither massless vector state nor massless tensor state are present. The restoration of the photon and the graviton imposes specific forms of the noncommutativity parameter matrices, partially removes the mass degeneracy and gives new additional ones. In particular, for the [Formula: see text]-branes, one can have a tachyon free model with a photon state when more strict conditions on these parameters are imposed, while, the match level condition of the closed string model induces the reduction of the spectrum.

Sign in / Sign up

Export Citation Format

Share Document