On the Algebra of q-Deformed Pseudodifferential Operators

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 q-deformed pseudodifferential operators. We use this algebraic structure to study in particular q-Burgers and q-KdV differential operators by the Lax generating technique. We give q-deformed Lax equations as well as the report between these equations through the q-deformed Burgers-KdV mapping.


q-Pseudodifferential Operators
We start this part with defining the q-derivation.For it, we are going to introduce the general case to know the α-derivation that is defined by where the two functions f and g are polynomials in an indeterminant x and its inverse x −1 .In 1.1 , α is a linear mapping.An example of the α-derivation is given by Jackson's q-differential operator ∂ q , such as 1 ISRN High Energy Physics which gives the following form for 1.1 : The q-shift operator η q is given by η q f x f qx .1.4 One can define the commutation relation as follows: where the multiplication law "•" is The last equation are obtained by using the following relation: where ∂ −1 q is the formal inverse of ∂ q .We should note that η q does not commute with ∂ q , ∂ q η k q f q k η k q ∂ q f , k ∈ Z 1.8 or in the following general case: ∂ m q η k q f q k m η k q ∂ m q f , k,m∈ Z. 1.9 Note that 1.6 can be unified as follows: for all n.In the last equation, the q-binomials take the form and the q-numbers are given by where the convention is taken.We can write out several explicit forms of 1.10 for q-derivative ∂ n q and ∂ −n q n ≥ 0 as We also add that the residue of the symbol L x, ∂ q can be written as and its Tr-functional is

Algebraic Structure of q-PDO
Now let us introduce the q-pseudodifferential operators algebra q-PDO.The latter is characterized by the relation 1 : 1.17 We can noted this space in the following way qA ≡ q−ΨDO is seen as being the algebra of all local and nonlocal q-differential operators of arbitrary conformal spins and arbitrary degrees, this spaces can be seen as being the q-deformation of pseudodifferential algebra A that we saw in 2-11 .One may expand qA as where we have denoted by m, n the lowest and the highest degrees, respectively, and by s the conformal spin.To be explicit, consider the space qA m,n s of q-differential operators: The vector space qA m,n of q-differential operators with given degrees m, n but undefined spin 1.20 exhibits a Lie algebra's structure with respect to the Lie bracket for m ≤ n ≤ 1.
In fact, It's straightforward to check that the commutator of two operators of qA p,q s is an operator of conformal spin 2s and degrees p, 2q − 1 .Since the Lie bracket •, • acts as imposing the closure, one gets strong constraints on the spin s and the degrees parameters m, n , namely, From these equations, we learn in particular that the spaces qA m,n 0 , m ≤ n ≤ 1 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 qA m,n 0 , m ≤ n ≤ 1 as well as the vector space qA 0,1 0 are in fact subalgebra of the Lie algebra qA −∞,1 0 which can be decomposed as is nothing but the Lie algebra of Lorentz scalar pure q-pseudodifferential operators of higher degree n −1 and qA 0,1 0 is the central extension of the Lie algebra qA 1,1 0 of vector fields Diff S 1 : and where qA 0,0 0 A 0,0 0 is the one dimensional trivial ideal.The infinite dimensional huge space qA is the algebra of q-differential operators of arbitrary spins and arbitrary degrees.It's obtained from the space qA m,n by summing over all allowed degrees qA:

1.25
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 qA is that it can splits into six infinite subalgebras qA j and qA j− , j 0, ±1 related to each others by conjugation of the spin and degrees.Indeed given two integers m and n ≥ m, it is not difficult to see that the vector spaces qA m,n and qA −n−1,−m−1 are dual with respect to the pairing product •, • defined as where d α,β are q-differential operators with fixed degrees α, β; β ≥ α but arbitrary spin and where the residue operation res is defined as: This equation shows that the operation res exhibits a conformal spin Δ 1.Using the properties of this operation and the pairing product 1.26 , one can decompose qA as follows: The indices and − carried by qA and qA − refer to the positive local and negative nonlocal degrees respectively.On the other hand one can decomposes the space qA m,m k , k ≥ 0 as

1.36
The duality of these 6 3 × 2 subalgebras is described by the combined scalar product •, • built out of the product equation 1.26 and conformal spin pairing: as follows 2, 3 : ISRN High Energy Physics 7 with respect to this new product, qA , qA 0 , and qA − behave as the dual algebras of qA −− , qA 0− , and qA − , respectively, while qA 0− is just the algebra of Lorenz scalar pure q-pseudo operators.This algebra and its dual qA 0 , the space of Lorenz scalar local q-differential operators, are very special subalgebras as they are systematically used to construct new realizations of the w i -symmetry, i ≥ 2 by using scalar q-differential operators type 1.39 We note that the space qA is the algebra of local q-differential operators of positive definite spins and positive degrees.qA −− , however, is the Lie algebra of pure qpseudodifferential operators of negative degrees and spins.

q-Deformed Lax Generating Technique
The aim of this section is to present some results related to the Lax representation in its qdeformed 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 q-deformation framework.
We underline that the present formulation is based on the q-pseudo operators ∂ n q and ∂ −n q instead of the pseudo operators ∂ n and ∂ −n 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 q 1.
The basic idea of the Lax formulation consists first in considering a noncommutative integrable system which possesses the Lax representation: .1 and the associated pair of operators L, B are called the Lax qdifferential equation and the Lax pair, respectively.The q-differential operator L defines the integrable system which we should fix from the beginning.
Note that the sl n -KdV hierarchy in the q-deformed version is defined as: and the way with which ones to writes the Lax q-differential equation as in 2.1 is equivalent to the following equation: where the operator B is the analogue of L k/2 describing then an q-differential operator of conformal spin k.

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Now, let us apply the q-deformation Lax-pair generating technique.We need to find an appropriate operator B which satisfies 2.1 , for this we have to make some constraints on the operator B, namely, Ansatz for the operator B: with ∂ n q is the q-differential operator which acts on L m according to 1.10 and B is another operator of same conformal weight than B.Then, with this ansatz, the problem reduces to find the operator B.
To understand the situation, we will study two interesting examples to know q-KdV and q-Burgers equations.

q-Deformed Burgers Equations
The L-operator for the q-deformed Burgers equation is given by with Let's consider the constraint n 1 m, for the q-deformed Burgers operator B can be written, from the ansatz 2.4 , as follows: Simply algebraic computations give where u 1 ∂u 1 /∂t.Now, our goal is to extract, from 2.1 and 2.8 , the Lax equation called q-deformed Burgers or just q-Burgers equation.For this we will follow the following procedure: Ansatz for the operator B: B α∂ q β, 2.9 where α and β are arbitrary functions on u and its derivatives.one finds

2.10
While identifying the two equations 2.8 and 2.10 we finds with a and b are arbitrary real constant.Equation 2.11 is called q-deformed Burgers equation or q-Burgers equation.the characteristic of this equation is that it is linear for b 1 and that for q 1. i.e., η q u 1 u 1 we recover the same equation gotten in works 4, 5, 9

q-Deformed KdV Equations
In this second example, we go worked on an q-differentials operator of conformal weight 2, this operator is given by the KdV Lax operator L q-KdV ∂ 2 q u 2 .

2.13
We are going to follow the same method of the previous example, therefore the Ansatz for the operator B is and the associated Lax equation: after a calculation, one finds by the same way of the case of Burgers, we finds the following q-KdV equation: as for q 1, we finds the standard KdV 2.18

q-Deformed Burgers-KdV Mapping
In this section, we present an approach to define the correspondence between integrables systems q-deformed-type Burgers and integrables systems q-deformed-type KdV.such correspondence named q-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 q-differentials-operators-type sl n -KdV.
Let's consider the Burgers q-differential operator 2.5 : 2.19 and the KdV q-differential operator 2.13 : .

2.20
Proposition 2.1 q-deformed Miura transformation .If one considers the two previous q-differential operators, one can make the following decomposition: with v 1 −η q u 1 and u 2 ∂ q −η q u 1 − u 1 η q u 1 .This decomposition is called q-deformed Miura transformation.one can see this mapping under the following form:

2.22
Proposition 2.2.As basing on the conforms weights of the operators derivatives: ∂ t q-KdV 3 and ∂ t q-Burgers 2, one can make the following correspondence: where α and β are arbitrary real constants.
Proposition 2.3 Généralisation .Being given an q -deformed Burgers operator L q-Burgers and an q-deformed sl n -KdV operator of type: then we can make the following decomposition: where v i , i 1, . . ., n are the fields of conformal weight 1 and which can be written in functions of the fields u j , j 2, . . ., n and their q-derivatives.

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 u s x, t .In This paper, we have devoted to a brief account of the basic properties of the space of q-pseudo differential Lax operators in the bosonic case.Presently, we know that any qpseudodifferential operator is completely specified by a conformal spin s, s ∈ Z, two integers p, and q p n, n ≥ 0 defining the lowest and the highest degrees, respectively, and finally 1 q − p n 1 analytic fields u j z .We recall that the space qA of all local and nonlocal qpseudodifferential operators admits a Lie algebra's structure with respect to the commutator buildout of the Leibnitz product.Moreover, we find that A splits into 3 × 2 6 subalgebr as qA j and qA j− , j 0, ±1 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 q-deformed integrable systems and the associated Lax generating technique.Particular interest is devoted to the q-Burgers and the q-KdV systems and their underlying mapping.
k .1.31 qΣ m,m k − and qΣ m,m k denote the spaces of q-differential operators of negative and positive definite spin.They are read as qΣ mvector space of Lorenz scalar q-differential operators.Combining 1.28 -1.34 , one sees that qA decomposes into 6 3 × 2 subalgebras qA ⊕ j 0, ,− qA j ⊕ qA j−