scholarly journals P∞ Algebra of KP, Free Fermions and 2-Cocycle in the Lie Algebra of Pseudodifferential Operators

1997 ◽  
Vol 11 (26n27) ◽  
pp. 3159-3193 ◽  
Author(s):  
A. Yu. Orlov ◽  
P. Winternitz
Keyword(s):  
Lie Algebra ◽  
Pseudodifferential Operators ◽  
Virasoro Algebra ◽  
Symmetry Algebra ◽  
Lie Point Symmetries ◽  
Negative Part ◽  
Fermion Algebra ◽  
Kp Equations ◽  
Time Variables ◽  
Positive Half

The symmetry algebra P∞=W∞⊕ H ⊕ I∞ of integrable systems is defined. As an example the classical Sophus Lie point symmetries of all higher KP equations are obtained. It is shown that one ("positive") half of the point symmetries belongs to the W∞ symmetries while the other ("negative") part belongs to the I∞ ones. The corresponding action on the τ-function is obtained. A new embedding of the Virasoro algebra into gl(∞) describes conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle.

SOLITON EQUATIONS AND THE VIRASORO ALGEBRA

1989 ◽  
Vol 04 (02) ◽  
pp. 467-479 ◽  
Author(s):  
A. M. SEMIKHATOV
Keyword(s):  
Riemann Surface ◽  
Central Extension ◽  
Vector Fields ◽  
Fock Space ◽  
Pseudodifferential Operators ◽  
Virasoro Algebra ◽  
Space Representation ◽  
Kp Equations ◽  
Set Up ◽  
Kp Theory

Starting in the Grassmannian set-up with a Fock space representation of the Virasoro algebra related by the Krichever map to the action of Diff S1 on J differentials on a Riemann surface, we proceed to the Kadomtsev-Petviashvily (KP) equations and show the Diff S1 generators to become certain vector fields on a manifold [Formula: see text] of pseudodifferential operators arising in the KP theory. These vector fields furnish a representations with c = 0. We conjecture that constructing its central extension may involve the Gelfand-Dikii symplectic structures on [Formula: see text].

Derivations and Automorphism Group of Original Deformative Schrödinger-Virasoro Algebra

2015 ◽  
Vol 22 (03) ◽  
pp. 517-540 ◽  
Author(s):  
Qifen Jiang ◽  
Song Wang
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Direct Product ◽  
Virasoro Algebra ◽  
Witt Algebra ◽  
Tensor Density ◽  
Derivation Algebra

In this paper, we determine the derivation algebra and the automorphism group of the original deformative Schrödinger-Virasoro algebra, which is the semi-direct product Lie algebra of the Witt algebra and its tensor density module Ig(a,b).

scholarly journals RESIDUAL SYMMETRIES IN THE PRESENCE OF AN EM BACKGROUND

2003 ◽  
Vol 18 (09) ◽  
pp. 629-641 ◽  
Author(s):  
H. L. CARRION ◽  
M. ROJAS ◽  
F. TOPPAN
Keyword(s):  
Lie Algebra ◽  
Symmetry Algebra ◽  
Algebraic Model ◽  
Two Dimensional ◽  
Square Roots ◽  
Residual Symmetry ◽  
Solvable Lie Algebra ◽  
Poincaré Algebra ◽  
Model Independent ◽  
Symmetry Algebras

The symmetry algebra of a QFT in the presence of an external EM background (named "residual symmetry") is investigated within a Lie-algebraic, model-independent scheme. Some results previously encountered in the literature are extended here. In particular we compute the symmetry algebra for a constant EM background in D = 3 and D = 4 dimensions. In D = 3 dimensions the residual symmetry algebra, for generic values of the constant EM background, is isomorphic to [Formula: see text], with [Formula: see text] the centrally extended two-dimensional Poincaré algebra. In D = 4 dimension the generic residual symmetry algebra is given by a seven-dimensional solvable Lie algebra which is explicitly computed. Residual symmetry algebras are also computed for specific non-constant EM backgrounds and in the supersymmetric case for a constant EM background. The supersymmetry generators are given by the "square roots" of the deformed translations.

EXTENDED BOTT–VIRASORO ALGEBRA, SEMIDIRECT PRODUCT, ⋆-LIE ALGEBRA OF DIFFEOMORPHISM AND NONCOMMUTATIVE INTEGRABLE SYSTEMS

2009 ◽  
Vol 06 (04) ◽  
pp. 555-572
Author(s):  
PARTHA GUHA
Keyword(s):  
Lie Algebra ◽  
Semidirect Product ◽  
Virasoro Algebra ◽  
Lie Derivative ◽  
Coadjoint Representation ◽  
Kdv Equations ◽  
Two Component ◽  
Universal Extension ◽  
Product Algebra ◽  
Tensor Densities

We study noncommutative theory of a coadjoint representation of a universal extension of Vect (S1) ⋉ C∞(S1) algebra using the action of ⋆-deformed matrix Hill's operators Δ⋆ on the space of ⋆-deformed tensor densities. The centrally extended semidirect product algebra [Formula: see text] is a Lie algebra of extended semidirect product of the Bott–Virasoro group [Formula: see text]. The study of deformed diffeomorphisms, deformed semidirect product algebra and deformed Lie derivative action of Δ⋆ on ⋆ deformed tensor-densities on S1 allow us to construct noncommutative two component Korteweg–de Vries (KdV) equations, in particular, we derive the noncommutative Ito equation. This leads to a geometric formulation of ⋆-deformed quantization of the centrally extended semidirect product algebra [Formula: see text] and two component noncommutative KdV equations.

THE CONFORMAL BOOTSTRAP AND SUPER W ALGEBRAS

1992 ◽  
Vol 07 (03) ◽  
pp. 591-617 ◽  
Author(s):  
JOSÉ M. FIGUEROA-O'FARRILL ◽  
STANY SCHRANS
Keyword(s):  
Systematic Study ◽  
Minimal Model ◽  
Central Charge ◽  
Virasoro Algebra ◽  
Symmetry Algebra ◽  
Modular Invariant ◽  
Conformal Bootstrap ◽  
Nonlinear Algebra

We undertake a systematic study of the possible extensions of the N = 1 super Virasoro algebra by a superprimary field of spin [Formula: see text]. Besides new extensions, which exist only for specific values of the central charge, we find a new nonlinear algebra (super W2) generated by a spin 2 superprimary which is associative for all values of the central charge. Furthermore, the spin 3 extension is argued to be the symmetry algebra of the m = 6 super Virasoro unitary minimal model, by exhibiting the (A7, D4)-type modular invariant as diagonal in terms of extended characters.

scholarly journals On the Orbits of One Non-Solvable 5-Dimensional Lie Algebra

2019 ◽  
pp. 5-20
Author(s):  
Artem Atanov ◽  
Alexander Loboda
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Fourth Order ◽  
Algebraic Surface ◽  
Vector Fields ◽  
Symmetry Algebra ◽  
Basis Vector ◽  
Invariant Polynomial ◽  
Real Hypersurfaces ◽  
Symmetry Algebras

This paper studies holomorphic homogeneous real hypersurfaces in C3 associated with the unique non-solvable indecomposable 5-dimensional Lie algebra 𝑔5 (in accordance with Mubarakzyanov’s notation). Unlike many other 5-dimensional Lie algebras with “highly symmetric” orbits, non-degenerate orbits of 𝑔5 are “simply homogeneous”, i.e. their symmetry algebras are exactly 5-dimensional. All those orbits are equivalent (up to holomorphic equivalence) to the specific indefinite algebraic surface of the fourth order. The proofs of those statements involve the method of holomorphic realizations of abstract Lie algebras. We use the approach proposed by Beloshapka and Kossovskiy, which is based on the simultaneous simplification of several basis vector fields. Three auxiliary lemmas formulated in the text let us straighten two basis vector fields of 𝑔5 and significantly simplify the third field. There is a very important assumption which is used in our considerations: we suppose that all orbits of 𝑔5 are Levi non-degenerate. Using the method of holomorphic realizations, it is easy to show that one need only consider two sets of holomorphic vector fields associated with 𝑔5. We prove that only one of these sets leads to Levi non-degenerate orbits. Considering the commutation relations of 𝑔5, we obtain a simplified basis of vector fields and a corresponding integrable system of partial differential equations. Finally, we get the equation of the orbit (unique up to holomorphic transformations) (𝑣 − 𝑥2𝑦1)2 + 𝑦2 1𝑦2 2 = 𝑦1, which is the equation of the algebraic surface of the fourth order with the indefinite Levi form. Then we analyze the obtained equation using the method of Moser normal forms. Considering the holomorphic invariant polynomial of the fourth order corresponding to our equation, we can prove (using a number of results obtained by A.V. Loboda) that the upper bound of the dimension of maximal symmetry algebra associated with the obtained orbit is equal to 6. The holomorphic invariant polynomial mentioned above differs from the known invariant polynomials of Cartan’s and Winkelmann’s types corresponding to other hypersurfaces with 6- dimensional symmetry algebras.

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.

INFINITE-DIMENSIONAL GENERALIZATIONS OF SIMPLE LIE ALGEBRAS

1990 ◽  
Vol 05 (24) ◽  
pp. 1967-1977 ◽  
Author(s):  
E. S. FRADKIN ◽  
V. YA. LINETSKY
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Virasoro Algebra ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Higher Spin ◽  
Higher Spin Theory

Infinite-dimensional algebras associated with simple finite-dimensional Lie algebra g are considered. Higher-spin generalizations of sl(2) are studied in detail. Those of the Virasoro algebra are viewed as their "analytic continuations". Applications in higher-spin theory and in conformal QFT are discussed.

MODIFIED KORTEWEG-DE VRIES EQUATIONS ON EUCLIDEAN LIE ALGEBRAS

1989 ◽  
Vol 03 (06) ◽  
pp. 853-861 ◽  
Author(s):  
B.A. KUPERSHMIDT
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Virasoro Algebra ◽  
One Dimensional ◽  
Finite Dimensional ◽  
De Vries

For any finite-dimensional Euclidean Lie alegebra [Formula: see text], a commuting hierarchy of generalized modified Korteweg-de Vries equations is constructed, together with a nonabelian generalization of the classical Miura map. The classical situation is recovered for the case when [Formula: see text] is abelian one-dimensional. Localization of differential formulae yields a representation of the Virasoro algebra in terms of elements of the current Lie algebra associated to [Formula: see text].

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