ISRN High Energy PhysicsVolume 2012 (2012), Article ID 503621, 12 pagesdoi:10.5402/2012/503621

Research Article

On the Algebra of -Deformed Pseudodifferential Operators

Abderrahman EL Boukili and Moulay Brahim Sedra

Département de Physique, Laboratoire des Hautes Energies, Sciences de l'Ingénierie et Réacteurs (LHESIR), Faculté des Sciences, Université Ibn Tofail, Kénitra, Morocco

Received 14 December 2011; Accepted 4 January 2012

Academic Editor: A. Belhaj

Copyright © 2012 Abderrahman EL Boukili and Moulay Brahim Sedra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

While basing on the study that we we achieved on pseudodifferential operators in the works [arXiv:0708.4046 and hep-th/0610056 ], we interest in this paper to the construction of the algebra of -deformed pseudodifferential operators. We use this algebraic structure to study in particular -Burgers and -KdV differential operators by the Lax generating technique. We give -deformed Lax equations as well as the report between these equations through the -deformed Burgers-KdV mapping.

1. Basic Notions

1.1. -Pseudodifferential Operators

We start this part with defining the -derivation. For it, we are going to introduce the general case to know the -derivation that is defined by where the two functions and are polynomials in an indeterminant and its inverse .

In (1.1), is a linear mapping. An example of the -derivation is given by Jackson's -differential operator , such as [1] which gives the following form for (1.1):

The -shift operator is given by

One can define the commutation relation as follows: where the multiplication law “” is The last equation are obtained by using the following relation: where is the formal inverse of .

We should note that does not commute with , or in the following general case:

Note that (1.6) can be unified as follows: for all . In the last equation, the -binomials take the form and the -numbers are given by where the convention is taken.

We can write out several explicit forms of (1.10) for -derivative and as We also add that the residue of the symbol can be written as and its Tr-functional is

1.2. Algebraic Structure of -PDO

Now let us introduce the -pseudodifferential operators algebra -PDO. The latter is characterized by the relation [1]:

We can noted this space in the following way is seen as being the algebra of all local and nonlocal -differential operators of arbitrary conformal spins and arbitrary degrees, this spaces can be seen as being the -deformation of pseudodifferential algebra that we saw in [2–11]. One may expand as where we have denoted by the lowest and the highest degrees, respectively, and by the conformal spin. To be explicit, consider the space of -differential operators:

The vector space of -differential operators with given degrees but undefined spin exhibits a Lie algebra's structure with respect to the Lie bracket for .

In fact, It's straightforward to check that the commutator of two operators of is an operator of conformal spin and degrees . Since the Lie bracket acts as imposing the closure, one gets strong constraints on the spin and the degrees parameters , namely, From these equations, we learn in particular that the spaces admit a Lie algebra's structure with respect to the bracket (1.5) provided that the Jacobi identity is fulfilled. This can be ensured by showing that the Leibnitz product is associative.

The spaces as well as the vector space are in fact subalgebra of the Lie algebra which can be decomposed as is nothing but the Lie algebra of Lorentz scalar pure -pseudodifferential operators of higher degree and is the central extension of the Lie algebra of vector fields : and where is the one dimensional trivial ideal.

The infinite dimensional huge space is the algebra of -differential operators of arbitrary spins and arbitrary degrees. It’s obtained from the space by summing over all allowed degrees : This infinite dimensional space which is the combined conformal spin and degrees tensor algebra is closed under the Lie bracket without any constraint.

A remarkable property of is that it can splits into six infinite subalgebras and related to each others by conjugation of the spin and degrees. Indeed given two integers and , it is not difficult to see that the vector spaces and are dual with respect to the pairing product defined as where are -differential operators with fixed degrees but arbitrary spin and where the residue operation is defined as: This equation shows that the operation exhibits a conformal spin . Using the properties of this operation and the pairing product (1.26), one can decompose as follows: with The indices + and − carried by and refer to the positive (local) and negative (nonlocal) degrees respectively. On the other hand one can decomposes the space as and denote the spaces of -differential operators of negative and positive definite spin. They are read as is just the vector space of Lorenz scalar -differential operators. Combining (1.28)–(1.34), one sees that decomposes into subalgebras with The duality of these subalgebras is described by the combined scalar product built out of the product equation(1.26) and conformal spin pairing: as follows [2, 3]: with respect to this new product, , , and behave as the dual algebras of , , and , respectively, while is just the algebra of Lorenz scalar pure -pseudo operators. This algebra and its dual , the space of Lorenz scalar local -differential operators, are very special subalgebras as they are systematically used to construct new realizations of the -symmetry, by using scalar -differential operators type

We note that the space is the algebra of local -differential operators of positive definite spins and positive degrees. , however, is the Lie algebra of pure -pseudodifferential operators of negative degrees and spins.

2. -Deformed Lax Generating Technique

The aim of this section is to present some results related to the Lax representation in its -deformed version. Using the convention notations and the analysis presented previously, we perform consistent algebraic computations, based on the Pseudodifferential analysis, to derive explicit Lax pair operators of some integrable systems in the -deformation framework.

We underline that the present formulation is based on the (-pseudo) operators and instead of the (pseudo) operators and used in several works. We note also that the obtained results are shown to be compatible with the ones already established in literature [12–16] in the case of .

The basic idea of the Lax formulation consists first in considering a noncommutative integrable system which possesses the Lax representation: with et .

Equation (2.1) and the associated pair of operators are called the Lax -differential equation and the Lax pair, respectively. The -differential operator defines the integrable system which we should fix from the beginning.

Note that the -KdV hierarchy in the -deformed version is defined as: and the way with which ones to writes the Lax -differential equation as in (2.1) is equivalent to the following equation: where the operator is the analogue of describing then an -differential operator of conformal spin .

Now, let us apply the -deformation Lax-pair generating technique. We need to find an appropriate operator which satisfies (2.1), for this we have to make some constraints on the operator , namely,

Ansatz for the operator : with is the -differential operator which acts on according to (1.10) and is another operator of same conformal weight than . Then, with this ansatz, the problem reduces to find the operator .

To understand the situation, we will study two interesting examples to know and equations.

2.1. -Deformed Burgers Equations

The -operator for the -deformed Burgers equation is given by with Let's consider the constraint , for the -deformed Burgers operator can be written, from the ansatz (2.4), as follows: Simply algebraic computations give where .

Now, our goal is to extract, from (2.1) and (2.8), the Lax equation called -deformed Burgers or just -Burgers equation. For this we will follow the following procedure:

Ansatz for the operator : where and are arbitrary functions on and its derivatives. one finds While identifying the two equations (2.8) and (2.10) we finds with and are arbitrary real constant.

Equation (2.11) is called -deformed Burgers equation or -Burgers equation. the characteristic of this equation is that it is linear for and that for . we recover the same equation gotten in works [4, 5, 9]

2.2. -Deformed KdV Equations

In this second example, we go worked on an -differentials operator of conformal weight 2, this operator is given by the KdV Lax operator

We are going to follow the same method of the previous example, therefore the Ansatz for the operator is and the associated Lax equation: after a calculation, one finds by the same way of the case of , we finds the following -KdV equation: as for , we finds the standard

2.3. -Deformed Burgers-KdV Mapping

In this section, we present an approach to define the correspondence between integrables systems -deformed-type Burgers and integrables systems -deformed-type KdV. such correspondence named -deformed Burgers-KdV mapping that is considered like a generalization of the Burgers-KdV mapping studied in works [7, 8, 11, 17].

We illustrate this idea with the example of KdV and Burgers equation and then we are going to make a generalization for cameraman -differentials-operators-type -KdV.

Let's consider the Burgers -differential operator (2.5): and the KdV -differential operator (2.13):

Proposition 2.1 (-deformed Miura transformation). If one considers the two previous operators, one can make the following decomposition: with and . This decomposition is called -deformed Miura transformation. one can see this mapping under the following form:

Proposition 2.2. As basing on the conforms weights of the operators derivatives: and , one can make the following correspondence: where and are arbitrary real constants.

Proposition 2.3 (Généralisation). Being given an -deformed Burgers operator and an -deformed -KdV operator of type: then we can make the following decomposition: where are the fields of conformal weight 1 and which can be written in functions of the fields and their -derivatives.

3. Conclusion

The importance of the theory of pseudodifferential operators in the study of nonlinear integrable systems is point out. Principally, the algebra of nonlinear (local and nonlocal) differential operators acte on the ring of analytic functions .

In This paper, we have devoted to a brief account of the basic properties of the space of -pseudo differential Lax operators in the bosonic case. Presently, we know that any -pseudodifferential operator is completely specified by a conformal spin , two integers , and defining the lowest and the highest degrees, respectively, and finally analytic fields . We recall that the space of all local and nonlocal -pseudodifferential operators admits a Lie algebra's structure with respect to the commutator buildout of the Leibnitz product. Moreover, we find that A splits into subalgebr as and related to each others by two types of conjugations, namely, the spin.

Finally, we have focused in this work to present the basics steps towards constructing the -deformed integrable systems and the associated Lax generating technique. Particular interest is devoted to the -Burgers and the -KdV systems and their underlying mapping.

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