Nuclear Realization of Virasoro–Zamolodchikov-w∞ ⋆-Lie Algebras Through the Renormalized Higher Powers of Quantum Meixner White Noise

2010 ◽  
Vol 17 (02) ◽  
pp. 135-160 ◽  
Author(s):  
Abdessatar Barhoumi ◽  
Anis Riahi
Keyword(s):  
White Noise ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Generalized Functions ◽  
Fock Spaces ◽  
Young Function ◽  
General Characterization ◽  
Nuclear Spaces ◽  
Characterization Theorems

By using an appropriate one-mode type interacting Fock spaces, [Formula: see text], introduced in [1], we define a nuclear triple [Formula: see text] of test and generalized functions, with θ being a suitable Young function. Moreover, we prove general characterization theorems for the fundamental nuclear spaces. For the applications, we introduce new renormalized products for the generators of the renormalized higher powers of white noise ⋆-Lie algebra and the Virasoro-Zamolodchikov-w∞ ⋆-Lie algebra. Then we show that these new renormalized products lead to nuclear realizations of these Lie algebras in terms of quantum Meixner white noise operators.

scholarly journals SYMPLECTIC AND ORTHOGONAL LIE ALGEBRA TECHNOLOGY FOR BOSONIC AND FERMIONIC OSCILLATOR MODELS OF INTEGRABLE SYSTEMS

2001 ◽  
Vol 16 (07) ◽  
pp. 1199-1225 ◽  
Author(s):  
A. J. MACFARLANE ◽  
HENDRYK PFEIFFER ◽  
F. WAGNER
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bethe Ansatz ◽  
Integrable Systems ◽  
Fock Spaces ◽  
Efficient Treatment ◽  
Orthogonal Lie Algebra ◽  
Integral Systems ◽  
Quantum Integrable Models ◽  
Invariant Tensors

To provide tools, especially L-operators, for use in studies of rational Yang–Baxter algebras and quantum integrable models when the Lie algebras so (N)(bn, dn) or sp (2n)(cn) are the invariance algebras of their R matrices, this paper develops a presentation of these Lie algebras convenient for the context, and derives many properties of the matrices of their defining representations and of the ad-invariant tensors that enter their multiplication laws. Metaplectic-type representations of sp (2n) and so (N) on bosonic and on fermionic Fock spaces respectively are constructed. Concise general expressions (see (5.2) and (5.5) below) for their L-operators are obtained, and used to derive simple formulas for the T operators of the rational RTT algebra of the associated integral systems, thereby enabling their efficient treatment by means of the algebraic Bethe ansatz.

scholarly journals MAD SUBALGEBRAS AND LIE SUBALGEBRAS OF AN ENVELOPING ALGEBRA

2010 ◽  
Vol 82 (3) ◽  
pp. 401-423
Author(s):  
XIN TANG
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Two Dimensional ◽  
Base Field ◽  
Inner Derivation ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Characterization Theorems

AbstractLet 𝒰(𝔯(1)) denote the enveloping algebra of the two-dimensional nonabelian Lie algebra 𝔯(1) over a base field 𝕂. We study the maximal abelian ad-nilpotent (mad) associative subalgebras and finite-dimensional Lie subalgebras of 𝒰(𝔯(1)). We first prove that the set of noncentral elements of 𝒰(𝔯(1)) admits the Dixmier partition, 𝒰(𝔯(1))−𝕂=⋃ 5i=1Δi, and establish characterization theorems for elements in Δi, i=1,3,4. Then we determine the elements in Δi, i=1,3 , and describe the eigenvalues for the inner derivation ad Bx,x∈Δi, i=3,4 . We also derive other useful results for elements in Δi, i=2,3,4,5 . As an application, we find all framed mad subalgebras of 𝒰(𝔯(1)) and determine all finite-dimensional nonabelian Lie algebras that can be realized as Lie subalgebras of 𝒰(𝔯(1)) . We also study the realizations of the Lie algebra 𝔯(1) in 𝒰(𝔯(1)) in detail.

STOCHASTIC HEAT EQUATION ON ALGEBRA OF GENERALIZED FUNCTIONS

2012 ◽  
Vol 15 (04) ◽  
pp. 1250026 ◽  
Author(s):  
ABDESSATAR BARHOUMI ◽  
HABIB OUERDIANE ◽  
HAFEDH RGUIGUI
Keyword(s):  
Heat Equation ◽  
Diffusion Process ◽  
White Noise ◽  
Lie Algebra ◽  
Stochastic Equations ◽  
Generalized Functions ◽  
Stochastic Heat Equation ◽  
Renormalization Procedure ◽  
Poisson Equations

The main objective of this paper is to investigate an extension [Formula: see text] of the "Volterra-Gross" Laplacian on nuclear algebra of generalized functions. In so doing, without using the renormalization procedure, this extension provides a continuous nuclear realization of the square white noise Lie algebra obtained by Accardi–Franz–Skeide in Ref. 2. An extended-Gross diffusion process driven by a class of Itô stochastic equations is studied, and solution of the related Poisson equations is derived in terms of a suitable λ-potential.

Higher powers of analytical operators and associated ∗-Lie algebras

2016 ◽  
Vol 19 (02) ◽  
pp. 1650013 ◽  
Author(s):  
Aymen Ettaieb ◽  
Narjess Turki Khalifa ◽  
Habib Ouerdiane ◽  
Hafedh Rguigui
Keyword(s):  
White Noise ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Entire Functions ◽  
New Product ◽  
Product Formula ◽  
Hermite Function ◽  
Test Functions ◽  
Infinite Dimensional ◽  
New Family

We introduce a new product of two test functions denoted by [Formula: see text] (where [Formula: see text] and [Formula: see text] in the Schwartz space [Formula: see text]). Based on the space of entire functions with [Formula: see text]-exponential growth of minimal type, we define a new family of infinite dimensional analytical operators using the holomorphic derivative and its adjoint. Using this new product [Formula: see text], such operators give us a new representation of the centerless Virasoro–Zamolodchikov-[Formula: see text]∗-Lie algebras (in particular the Witt algebra) by using analytical renormalization conditions and by taking the test function [Formula: see text] as any Hermite function. Replacing the classical pointwise product [Formula: see text] of two test functions [Formula: see text] and [Formula: see text] by [Formula: see text], we prove the existence of new ∗-Lie algebras as counterpart of the classical powers of white noise ∗-Lie algebra, the renormalized higher powers of white noise (RHPWN) ∗-Lie algebra and the second quantized centerless Virasoro–Zamolodchikov-[Formula: see text]∗-Lie algebra.

COHOMOLOGY OF THE VIRASORO–ZAMOLODCHIKOV AND RENORMALIZED HIGHER POWERS OF WHITE NOISE *-LIE ALGEBRAS

2009 ◽  
Vol 12 (02) ◽  
pp. 193-212 ◽  
Author(s):  
LUIGI ACCARDI ◽  
ANDREAS BOUKAS
Keyword(s):  
White Noise ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Unitary Representations ◽  
Central Extensions

We prove the triviality of the second cohomology group of the Virasoro–Zamolodchikov and Renormalized Higher Powers of White Noise *-Lie algebras. It follows that these algebras admit only trivial central extensions. We also prove that the Heisenberg–Weyl *-Lie algebra admits nontrivial central extensions which are parametrized in a 1-to-1 way by ℂ\{0}. Explicit unitary *-representations of these extensions and their implications for our renormalization program are discussed in Ref. 8.

UNITARY REPRESENTATIONS OF THE WITT AND sl(2, ℝ)-ALGEBRAS THROUGH RENORMALIZED POWERS OF THE QUANTUM PASCAL WHITE NOISE

2008 ◽  
Vol 11 (03) ◽  
pp. 323-350 ◽  
Author(s):  
ABDESSATAR BARHOUMI ◽  
HABIB OUERDIANE ◽  
ANIS RIAHI
Keyword(s):  
White Noise ◽  
Test Functions ◽  
Young Function ◽  
Witt Algebra ◽  
Renormalization Procedure ◽  
Analytical Functions ◽  
Space Of Distributions ◽  
Chaos Decomposition ◽  
Noise Test ◽  
Characterization Theorems

By using an appropriate space of distributions, [Formula: see text], we derive the chaos decomposition property of the Hilbert space of quadratic integrable functionals with respect to the Pascal white noise measure ΛNB. The constructed decomposition is used to define a nuclear triple [Formula: see text] of test and generalized functions, where θ is a Young function satisfying some suitable conditions. A general characterization theorems are proven for the Pascal white noise distributions, white noise test functions and white noise operators in terms of analytical functions with growth condition of exponential type. By using appropriate renormalization procedure, we obtain the representation of the square of white noise obtained by Accardi–Franz–Skeide in Ref. 5. Finally, we investigate the main aim of this paper which is to give unitary equivalent representations of the Witt algebra in the basis of Pascal white noise theory.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

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