On the Kadomtsev-Petviashvili hierarchy in an extended class of formal pseudo-differential operators
aa r X i v : . [ m a t h - ph ] J a n ON THE KADOMTSEV-PETVIASHVILI HIERARCHY IN ANEXTENDED CLASS OF FORMAL PSEUDO-DIFFERENTIALOPERATORS
JEAN-PIERRE MAGNOT AND VLADIMIR ROUBTSOV , , Abstract.
We study the existence and uniqueness of the Kadomtsev-Petviashvili(KP) hierarchy solutions in the algebra of F Cl ( S , K n ) of formal classicalpseudo-differential operators. The classical algebra Ψ DO ( S , K n ) where theKP hierarchy is well-known appears as a subalgebra of F Cl ( S , K n ) . We inves-tigate algebraic properties of F Cl ( S , K n ) such as splittings, r-matrices, exten-sion of the Gelfand-Dickii bracket, almost complex structures. Then, we provethe existence and uniqueness of the KP hierarchy solutions in F Cl ( S , K n ) with respect to extended classes of initial values. Finally, we extend this KPhierarchy to complex order formal pseudo-differential operators and we de-scribe their Hamiltonian structures similarly to previously known formal case.. Keywords:
Formal pseudo-differential operators, Kadomtsev-Petviashvili hierar-chy, almost complex structure, almost quaternionic structure.
MSC (2020):
Introduction
In the classical theory of the Kadomtsev-Petviashvili (KP) hierarchy, the con-sidering algebra of pseudo-operators is Ψ DO ( S , K ) = C ∞ ( S , K )(( ∂ − )) where ∂ is a derivation and K = R , C or H . Classically, ∂ = ddx , x ∈ S . It is well-known that this KP hierarchy is an integrable system, with existence and uniquenessof solutions with respect to a fixed initial value, ( see e.g. [6] for a classical treatise).There exists various generalizations, or deformations, of the KP hierarchy whichalmost all satisfy the formal integrability condition, and solutions satisfy propertiessimilar to the properties of the solutions of the (classical) KP hierarchy. Recently,well-posedness have been stated for these equations [10, 25]. Classical algebraicsettings that arise in the theory of the KP hierarchy will be reviewed in section 1.3,in section 1.4 and in section 1.7.Pseudo-differential operators appear also in some contexts other than the theoryof integrable systems. In general, larger classes of such operators are studied, see e.g.[13, 30, 32, 33], starting from non-formal operators, i.e. operators acting on spacesof sections of a vector bundle. These non-formal operators, in particular classical pseudo-differential operators, have their own applications and one can build fromthem spaces of formal classical operators. The algebra of operators that we intendto use in this paper is the algebra of formal classical pseudo-differential operators F Cl ( S , K n ) that are obtained from classical pseudo-differential operators acting AND VLADIMIR ROUBTSOV , , on smooth sections of the trivial vector bundle S × K n over S , for K = C or H , see e.g. [13, 30]. In this algebra, it is possible to define functions of an ellipticpositive operator that satisfy mild properties of the spectrum using a Cauchy-likeformula [30, 32, 33]. In particular the square root of the Laplacian | D | = ∆ / iswell-defined, as well as the sign of the Dirac operator D = i ddx defined by ǫ ( D ) = D | D | − = | D | − D. This operator is not in Ψ DO ( S , K n ) . In fact, the algebra Ψ DO ( S , K ) is the formalpart of the so-called even-even class of (non-formal) classical pseudo-differenbtialoperators first defined, to our knowledge, by Kontsevich and Vishik [16, 17] andnamed as even-even class operators in [30, 32], mostly motivated by problems aboutrenormalized determinants. As a consequence, Ψ DO ( S , K ) is a subalgebra of F Cl ( S , K n ) which is noted in [30, 32] as F Cl ee ( S , K n ) . The necessary propertiesof these pseudo-differential operator algebras, both formal and non-formal, will bereviewed in section 1.1. The key properties of ǫ ( D ) that we use in our constructionsare: • the formal operator ǫ ( D ) ∈ F Cl ( S , K n ) commutes with any formal oper-ator A ∈ F Cl ( S , K n ) , • ǫ ( D ) = Id • the composition on the left A ǫ ( D ) ◦ A is an endomorphism of the alge-bra F Cl ( S , K n ) , which restricts to a bijiective map from Ψ DO ( S , K n ) = F Cl ee ( S , K n ) to an algebraic complement in F Cl ( S , K n ) noted as F Cl eo ( S , K n ) following the terminology of [32] • the restriction of the Wodzicki residue to Ψ DO ( S , K n ) = F Cl ee ( S , K n ) ,which is similar to but not equal to the Adler functional, is vanishing.Our first remarks are the following: • The space F Cl ( S , K n ) splits in various ways: one is derived from thesplitting of T ∗ S − S into two connected components (section 1.1.2), thesplitting with respect to Ψ DO ( S , K n ) as a subalgebra (section 1.1.3),and the extension of the splitting related to the classical Manin triple on Ψ DO ( S , K n ) to F Cl ( S , K n ) (section 1.5.1) . • The operator ǫ ( D ) is in the center of F Cl ( S , K n ) . It generates then apolarized Lie bracket using it as a r − matrix (section 1.6) and an integrablealmost complex structure on F Cl ( S , K n ) . These technical features enables us to state the announced main results of thispaper: existence and uniqueness of solutions of the KP hierarchy with various initialconditions (section 3.1) and KP hierarchy with complex powers (section 3.2).The paper is organized as follows:Section 1 is devoted to technical preliminaries: we remind and review some op-erator algebras, Poisson structures and Manin pairs. We give an overview of theclassical method for solving the KP hierarchy. New results of this Section areconcentrated in section 1.2 where formal operators of complex order that general-ize operators in F CL ( S , K n ) , extending the definitions present in [11], [19], aredescribed. In section 1.5, we explore some Manin pairs on F Cl ( S K n ) , and insection 1.6 we present some polarized brackets, inherited from the richer structureof F Cl ( S K n ) .Section 2.1 is focused on the comparison of F Cl ( S K n ) with Ψ DO ( S , K n ) . First, we develop various injections of Ψ DO ( S , K n ) in F Cl ( S K n ) , beyond the EALIZATION OF QUATERNIONIC KP HIERARCHY 3 standard one described in section 1. Second, we describe three almost complexstructures on F Cl ( S K n ) J , J and J such that each couple ( J , J ) , ( J , J ) and ( J , J ) form an almost quaternionic structure on F Cl ( S K n ) . We prove theintegrability of J , derived from iǫ ( D ) , and the non-integrability of the two others J and J . Section 3 deals with various type of initial values for the KP system, which arederived from the various injections of Ψ DO ( S , K n ) in F Cl ( S , K n ) , and ends upwith a generalization to the KP hierarchy with operators of complex order. As itwas announced, the existence and uniqueness of the solutions, depending on theinitial value, is stated. We make few short remarks about well-posedness.The final part of the paper extends the classical Hamiltoinian formulations ofthe KP hierarchy from Ψ DO ( S , K n ) to F CL ( S , K n ) , using a generalized Adler-Gelfand-Dickii construction.All technical and routine proofs are gathered and organized in the Appendix.0.1. Acknowledgements.
This research of both authors was supported by LAREMAUMR 6093 du CNRS. V.R. was partly supported by the project IPaDEGAN (H2020-MSCA-RISE-2017), Grant Number 778010, and by the Russian Foundation forBasic Research under the Grants RFBR.0.2.
Conflict of Interest.
The authors declare that they have no conflicts ofinterest. 1.
Technical preliminaries
Preliminaries on pseudo-differential operators.
Description.
We shall start with a description of (non-formal!) pseudo-differential operator groups and algebras which we consider in this work. Through-out this section E denotes a complex finite-dimensional vector bundle over S . Weshall specialize below to the case E = S × V in which V is a n − dimensional vectorspace. The following definition appears in [2, Section 2.1]. Definition 1.1.
The graded algebra of differential operators acting on the space ofsmooth sections C ∞ ( S , E ) is the algebra DO ( E ) generated by: • elements of End ( E ) , the group of smooth maps E → E leaving each fibreglobally invariant and which restrict to linear maps on each fibre. This group actson sections of E via (matrix) multiplication; • covariant derivation operators ∇ X : g ∈ C ∞ ( S , E )
7→ ∇ X g where ∇ is a smooth connection on E and X is a smooth vector field on S . We assign as usual the order to smooth function multiplication operators.The derivation operators and vector fields have the order 1. A differential op-erator of order k has the form P ( u )( x ) = P p i ··· i r ∇ x i · · · ∇ x ir u ( x ) , r ≤ k , In local coordinates (the coefficients p i ··· i r can be matrix-valued). We denote by DO k ( S ) , k ≥ , the differential operators of order less or equal than k . The alge-bra DO ( E ) is filtered by the order. It is a subalgebra of the algebra of classicalpseudo-differential operators Cl ( S , V ) that we describe shortly hereafter, focusing JEAN-PIERRE MAGNOT AND VLADIMIR ROUBTSOV , , on its necessary aspects. This is an algebra that contains, for example, the squareroot of the Laplacian(1) | D | = ∆ / = Z Γ λ / (∆ − λId ) − dλ, where ∆ = − d dx is the positive Laplacian and Γ is a contour around the spec-trum of the Laplacian, see e.g. [33, 30] for an exposition on contour integrals ofpseudo-differential operators. Cl ( S , V ) contains also the inverse of Id + ∆ , and allsmoothing operators on L ( S , V ) . Among smoothing operators one can find theheat operator e − ∆ = Z Γ e − λ (∆ − λId ) − dλ. pseudo-differential operators (maybe non-scalar) are linear operators acting on C ∞ ( S , V ) which reads locally as A ( f ) = Z e ix.ξ σ ( x, ξ ) ˆ f ( ξ ) dξ where σ ∈ C ∞ ( T ∗ S , M n ( C )) satisfying additional estimates on its partial deriva-tives and ˆ f means the Fourier transform of f . Basic facts on pseudo-differentialoperators defined on a vector bundle E → S can be found e.g. in [13]. Remark 1.2.
Since V is finite dimensional, there exists n ∈ N ∗ such that V ∼ C n . Through this identification, a pseudo-differential operator A ∈ Cl ( S , V ) can beidentified with a matrix ( A i,j ) ( i,j ) ∈ N n with coefficients A i,j ∈ Cl ( S , C ) . In other words, the identification V ∼ C n that we fix induces the isomorphism ofalgebras Cl ( S , V ) ∼ M n ( Cl ( S , C )) . This identification will remain true and useful in the successive constructions below,and will be recalled if appropriate. When it will not carry any ambiguity, we willuse the notation DO ( S ) , Cl ( S ) , etc. instead of DO ( S , C ) , Cl ( S , C ) , etc. foroperators acting on the space of smooth functions from S to C . Pseudo-differential operators can be also described by their kernel K ( x, y ) = Z R e i ( x − y ) ξ σ ( x, ξ ) dξ which is off-diagonal smooth. Pseudo-differential operators with infinitely smoothkernel (or "smoothing" operators), i.e. that are maps: L → C ∞ form a two-sided ideal that we note by Cl −∞ ( S , V ) . Their symbols are those which are in theSchwartz space S ( T ∗ S , M n ( C )) . The quotient F Cl ( S , V ) = Cl ( S , V ) /Cl −∞ ( S , V ) of the algebra of pseudo-differential operators by Cl −∞ ( S , V ) forms the algebra offormal pseudo-differential operators. Another algebra, which is actually known asa subalgebra of F Cl ( S , V ) following [26], is also called algebra of formal pseudo-differential operators. This algebra is generated by formal Laurent series Ψ DO ( S , V ) = C ∞ ( S , V )(( ∂ − )) = [ d ∈ Z X k ≤ d a k ∂ k EALIZATION OF QUATERNIONIC KP HIERARCHY 5 where each a k ∈ C ∞ ( S , M n ( C )) and ∂ = ddx . Let us precise hereafter a short butcomplete description of basic correspondence between Ψ DO ( S , V ) and F Cl ( S , V ) .Symbols σ project to formal symbols and there is an isomorphism between for-mal pseudo-differntial operators and formal symbols. A detailed study can be foundin [7, Tome VII]. Classical pseudo-differential operators are operators A which as-sociated formal symbol σ ( A ) reads as an asymptotic expansion σ ( A )( x, ξ ) ∼ X k ∈ Z ,k ≤ o σ k ( A )( x, ξ ) where the partial symbol of order k σ k ( A ) : ( x, ξ ) ∈ T ∗ S \ S σ k ( A )( x, ξ ) ∈ M n ( C ) is k − positively homogeneous in the ξ − variable, smooth on T ∗ S \ S = { ( x, ξ ) ∈ T ∗ S | ξ = 0 } and such that d ∈ Z is the order of the operator A. The order ofa smoothing operator we put equal to −∞ and the formal symbol of a smoothingoperator is . The set F Cl ( S , V ) is not the same as the space of formal operators Ψ DO ( S , V ) which naturally arises in the algebraic theory of PDEs, see e.g. [18] for an overview,but here the partial symbols σ k ( A ) of A ∈ Ψ DO ( S , V ) are k − homogeneous. Bythe way one only has Ψ DO ( S , V ) ⊂ F Cl ( S , V ) . Following the remarks givenin [26], Ψ DO ( S , V ) correspond to even-even class formal pseudo-differentialoperators that we describe in section 1.1.3. Two approaches for a global symboliccalculus of pseudo-differential operators have been described in [4, 34]. It is shown inthese papers how the geometry of the base manifold M furnishes an obstruction togeneralizing local formulas of of symbol composition and inversion; we do not recallthese formulas here since they are not involved in our computations. We assumehenceforth (following e.g. [24], along the lines of the more general description of[13]), that S is equipped with charts such that the changes of coordinates aretranslations. Under these assumptions, σ ( A ◦ B ) ∼ X α ∈ N ( − i ) α α ! D αξ σ ( A ) D αx σ ( B ) , ∀ A, B ∈ Cl ( S , V ) , and specializing to partial symbols: ∀ k ∈ Z , σ ( A ◦ B ) k = X α ∈ N X m + n − α = k ( − i ) α α ! D αξ σ m ( A ) D αx σ n ( B ) . The composition σ ( A ◦ B ) for A, B ∈ Ψ DO ( S , V ) ⊂ F Cl ( S , V ) gives rise to a(unitary) associative algebra structure on Ψ DO ( S , V ) and we shall write in thiscase (by abuse of notation)(2) A ◦ B = X α ∈ N ( − i ) α α ! D αξ AD αx B Remark 1.3.
In such an "operator product" we shall always suppose so called"Wick order" which means that we write functions on C ∞ ( S ) on (or "in frontof") left-hand side of all degrees of D. Notations.
We shall denote note by Cl d ( S , V ) the vector space of classicalpseudo-differential operators of order ≤ d . We also denote by Cl ∗ ( S , V ) the group JEAN-PIERRE MAGNOT AND VLADIMIR ROUBTSOV , , of invertible in Cl ( S , V ) operators. We denote the sets of formal operators addingthe script F . The algebra of formal pseudo-differential operators, identified wihformal symbols, is noted by F Cl ( S , V ) , and its group of invertible element is F Cl ∗ ( S , V ) , while formal pseudo-differential operators of order less or equal to d ∈ Z is noted by F Cl d ( S , V ) . Remark 1.4.
Through identification of F Cl ( S , V ) with the corresponding spaceof formal symbols, the space F Cl ( S , V ) is equipped with the natural locally convextopology inherited from the space of formal symbols. A formal symbol σ k is a smoothfunction in C ∞ ( T ∗ S \ S , M n ( C )) which is k − homogeneous (for k > ), and hencewith an element of C ∞ ( S , M n ( C )) evaluating σ k at ξ = 1 and ξ = − . Identifyting Cl d ( S , V ) with Y k ≤ d C ∞ ( S , M n ( C )) , the vector space Cl d ( S , V ) is a Fréchet space, and hence Cl ( S , V ) = ∪ d ∈ Z Cl d ( S , V ) is a locally convex topological algebra.We have to precise that the classical topology on non-formal classical pseudo-differential operators Cl ( S , V ) is finer than the one obtained by pull-back from F Cl ( S , V ) . A “useful” topology on Cl ( S , V ) needs to ensure that partial sym-bols and off-diagonal smooth kernels converge. The topology on spaces of classicalpseudo differential operators has been described by Kontsevich and Vishik in [16] ; seealso [5, 30, 32] for descriptions. This is a Fréchet topology on each space Cl d ( S , E ) . However, passing to the quotients F Cl d ( S , E ) = Cl d ( S , E ) /Cl −∞ ( S , E ) , thepush-forward topology coincides with the topology of F Cl d ( S , V ) described at thebeginning of this remark. The splitting with induced by the connected components of T ∗ S \ S . . In thissection, we define two ideals of the algebra F Cl ( S , V ) , that we call F Cl + ( S , V ) and F Cl − ( S , V ) , such that F Cl ( S , V ) = F Cl + ( S , V ) ⊕ F Cl − ( S , V ) . Thisdecomposition is explicit in [15, section 4.4., p. 216], and we give an explicitdescription here following [21, 22]. Definition 1.5.
Let σ be a partial symbol of order o on E . Then, we define, for ( x, ξ ) ∈ T ∗ S \ S , σ + ( x, ξ ) = (cid:26) σ ( x, ξ ) if ξ > if ξ < and σ − ( x, ξ ) = (cid:26) if ξ > σ ( x, ξ ) if ξ < . We define p + ( σ ) = σ + and p − ( σ ) = σ − . The maps p + : F Cl ( S , V ) → F Cl ( S , V ) and p − : F Cl ( S , V ) → F Cl ( S , V ) are clearly smooth algebra morphisms (yet non-unital morphisms) that leave theorder invariant and are also projections (since multiplication on formal symbols isexpressed in terms of point-wise multiplication of tensors). Definition 1.6.
We define F Cl + ( S , V ) = Im ( p + ) = Ker ( p − ) and F Cl − ( S , V ) = Im ( p − ) = Ker ( p + ) . Since p + is a projection, we have the splitting F Cl ( S , V ) = F Cl + ( S , V ) ⊕ F Cl − ( S , V ) . EALIZATION OF QUATERNIONIC KP HIERARCHY 7
Let us give another characterization of p + and p − . The operator D = − i ddx splits C ∞ ( S , C n ) into three spaces : • its kernel E , built of constant maps • E + , the vector space spanned by eigenvectors related to positive eigenvalues • E − , the vector space spanned by eigenvectors related to negative eigenval-ues.The L − orthogonal projection on E is a smoothing operator, which has null formalsymbol. By the way, concentrating our attention on thr formal symbol of operators,we can ignore this projection and hence we work on E + ⊕ E − . The followingelementary result will be useful for the sequel. Lemma 1.7. [21, 22] • σ ( D ) = ξ, σ ( | D | ) = | ξ |• σ ( ǫ ) = ξ | ξ | , where ǫ = D | D | − = | D | − D is the sign of D. • Let p E + (resp. p E − ) be the projection on E + (resp. E − ), then σ ( p E + ) = ( Id + ξ | ξ | ) and σ ( p E − ) = ( Id − ξ | ξ | ) . Let us now give an easy but very useful lemma:
Lemma 1.8. [21]
Let f : R ∗ → V be a 0-positively homogeneous function withvalues in a topological vector space V . Then, for any n ∈ N ∗ , f ( n ) = 0 where f ( n ) denotes the n-th derivative of f . From this, we have the following result.
Proposition 1.9. [21, 22]
Let A ∈ F Cl ( S , V ) . p + ( A ) = σ ( p E + ) ◦ A = A ◦ σ ( p E + ) and p − ( A ) = σ ( p E − ) ◦ A = A ◦ σ ( p E − ) . Notation.
For shorter notations, we note by A ± = p ± ( A ) the formal operatorsdefined from another viewpoint by σ ( A + )( x, ξ ) ( resp. σ ( A − )( x, ξ )) = (cid:26) σ ( A )( x, ξ ) if ξ > resp. ξ < if ξ < resp. ξ > The “odd-even” splitting.
We note by σ ( A )( x, ξ ) the total formal symbol of A ∈ F Cl ( S , V ) . The following proposition is trivial:
Proposition 1.10.
Let φ : F Cl ( S , V ) → F Cl ( S , V ) defined by φ ( A ) = 12 X k ∈ Z σ k ( A )( x, ξ ) − ( − k σ k ( A )( x, − ξ ) . This map is smooth, and Ψ DO ( S , V ) = F Cl ee ( S , V ) = Ker ( φ ) . Following [32], one can define even-odd class pseudo-differential operators F Cl eo ( S , V ) = ( A ∈ F Cl ( S , V ) | X k ∈ Z σ k ( A )( x, ξ ) + ( − k σ k ( A )( x, − ξ ) = 0 ) . Remark 1.11.
This terminology is inherited from [32] . This reference is mostlyconcerned with non-formal operators. We have also to mention that the class ofnon formal even-even pseudo-differential operators was first described in [16, 17] .In these two references, even-even class pseudo-differential operators are called oddclass pseudo-differential operators. By the way, following the terminology of [16, 17]
JEAN-PIERRE MAGNOT AND VLADIMIR ROUBTSOV , , even-odd class pseudo-differential operators should be called even class. In this paperwe prefer to fit with the terminology given in the textbooks [30, 32] even if the initialterminology given in [16, 17] and its natural extension would appear more naturalto us. Proposition 1.12. φ is a projection and F Cl eo ( S , V ) = Imφ.
By the way, we also have F Cl ( S , V ) = F Cl ee ( S , V ) ⊕ F Cl eo ( S , V ) . We have the following composition rules for the class of a formal operator A ◦ B : A even-even class A even class B even-even class A ◦ B even-even class A ◦ B even-odd class B even-odd class A ◦ B even-odd class A ◦ B even-even class Example 1.13. ǫ ( D ) and | D | are even-odd class, while we already mentioned thatdifferential operators are even-even class. Remark 1.14.
The operator ǫ ( D ) satisfies the following properties: • Since ǫ ( D ) = Id, the left composition A ∈ F Cl ( S , V ) ǫ ( D ) ◦ A is aninvolution on F Cl ( S , V ) • Since ǫ ( D ) ∈ F Cl eo ( S , V ) , the restriction of ǫ ( D ) ◦ ( . ) to Ψ DO ( S , V ) = F Cl ee ( S , V ) is a bijection from F Cl ee ( S , V ) to F Cl eo ( S , V ) . One can also define the operator s on F Cl ( S , V ) which extends the operator s : T ∗ S → T ∗ S defined by s ( x, ξ ) = ( x, − ξ ) by s : X n σ n ( x, ξ ) X n ( − n σ n ( s ( x, ξ )) . This operator obviously satisfies s = Id, and we remark the following properties:
Proposition 1.15. • s (cid:0) F Cl ± ( S , V ) (cid:1) = F Cl ∓ ( S , V ) • F Cl ee ( S , V ) = Ker ( Id − s ) • F Cl eo ( S , V ) = Ker ( Id + s ) Remark 1.16.
One can consider also s ′ : P n σ n ( x, ξ ) P n σ n ( s ( x, ξ )) . We stillhave s ′ = Id, s ′ (cid:0) F Cl ± ( S , V ) (cid:1) = F Cl ∓ ( S , V ) but the two other properties arenot fulfilled. Under these properties, F Cl ee ( S , V ) and F Cl eo ( S , V ) appear respectively aseigen-spaces for the eigen values and − of the symmetry s , and hence an operator a ∈ F Cl ( S , V ) = F Cl ee ( S , V ) ⊕ F Cl eo ( S , V ) decomposes as a = a ee + a eo and s [ a, b ] = [ a, b ] ee − [ a, b ] eo = ([ a ee , b ee ] + [ a eo , b eo ]) − ([ a ee , b eo ] + [ a eo , b ee ]) . EALIZATION OF QUATERNIONIC KP HIERARCHY 9
Complex powers of a formal pseudo-differential operator.
Following[12] inspired by [33], this is possible to define the complex power of an elliptic formaloperator. Concerning formal operators, ellipticity is fully obtained by a conditionon the principal symbol of the operator. This provides the possibility, when thealgebra of functions R is e.g. a complete topological vector space with boundedaddition and multiplication laws, to define complex powers A α of a formal operator A for Re ( α ) < via contour integrals similar to (1) and then extend it to arbitrarycomplex powers. Beyond these technical problems, for any formal C − algebra offunctions R with differentiation ∂, it is possible to define the same complex powersof the Lax-type operators L ∈ Ψ DO ( R ) present in the KP hierarchy, along the linesof [19] and [11]. Let α ∈ C and let Ψ DO α ( R ) be the affine space of formal series ofthe form X k ∈ N a α − k ∂ α − k , formally defined as Ψ DO α ( R ) = Ψ DO ( R ) .∂ α . On the total spce of formal pseudo-diferential operators of complex order generated by the family (Ψ DO α ( R )) α ∈ C , thesame addition and multiplication rules as in Ψ DO ( R ) holds true and consistent.Let A ∈ Ψ DO α ( R ) with a α ∈ R ∗ + ⊂ R, one can define log( A ) ∈ α log a∂ + Ψ DO ( R ) such that exp (log( A )) = A by standard rules of formal series.Let L ∈ Ψ DO ( R ) with principal symbol ∂. We can then define the complexpower L α for α ∈ C ∗ , and following the notations of [11, 18], the affine space L = ∂ + Ψ DO ( R ) has an affine isomorphism, for α ∈ C ∗ , with L α = ∂ α + Ψ DO α − ( R ) through the identification L ∈ L 7→ L α = exp ( α log( L )) ∈ L α . From this con-struction on Ψ DO ( S , K ) , one can push forward complex powers on subalgebrasof F Cl ( S , K ) via the identifications already described. More precisely, one useheuristically the bijection Φ , : Ψ DO ( S , K ) → F Cl + ( S , K ) to define, for A = ddx + + P k ≤ a k ddx k + ∈ F Cl ( S , K ) , first the logarithm log A = log ddx + + X k ≤ a k ddx k + and the complex power A α = exp ( α log A ) which formal symbol vanishes for ξ < . Then we define F Cl α + ( S , K ) = F Cl ( S , K ) ddx α + and (after these constructions) Φ , extends naturally to a bijection from Ψ DO α ( S , K ) to F Cl α + ( S , K ) . The same construction holds to extend the identification of Ψ DO ( S , K ) with F Cl − ( S , K ) to complex powers Φ , : Ψ DO α ( S , K ) → F Cl α − ( S , K ) and de-fine F Cl α ( S , K ) = F Cl α + ( S , K ) ⊕ F Cl α − ( S , K ) = (Φ , × Φ , ) (cid:0) Ψ DO α ( S , K ) (cid:1) . One can also understand F Cl α ( S , K ) as F Cl α ( S , K ) = F Cl ( S , K ) | D | α where | D | α = ∆ α is defined via Seeley’s complex powers [33]. Alternatively, setting (cid:18) ddx (cid:19) α = (cid:18) ddx (cid:19) α + + (cid:18) ddx (cid:19) α − = iǫ ( D ) | D | α , AND VLADIMIR ROUBTSOV , , we get F Cl α ( S , K ) = F Cl ( S , K ) (cid:0) ddx (cid:1) α . These spaces of complex powers containthe projections on formal operators (up to smoothing oprators) of the classes ofpseudo-differential operators of complex order defined in [16, section 3].1.3.
Lie-algebraic digression.
Operator bialgebras and Manin pairs.
One can easily define a Lie algebrastructure by antysimmetrisation of the associative product [ A, B ] = A ◦ B − B ◦ A .We remark that the vector field Lie algebra Vect( S ) and its semi-direct productwith C ∞ ( S ) = C ∞ ( S , C ) is a natural Lie subalgebra of the differential operatorLie algebra DO ( S ) which is formed by the order 1 differential operators and theorder less or equal to 1. This remark can be also deduced from Definition 1.1by setting E = S × C , i.e. V = C . When V = C n with n ≥ , an operator X ∈ V ect ( S ) can be identified with the degree 1 differential opeartor X ⊗ Id C n ∈ DO ( S , V ) while order differential operators coincide with multplication operatorsin C ∞ ( S , M n ( C )) . We also have that C ∞ ( S , M n ( C )) ⋊ V ect ( S ) ⊂ DO ( S , V ) as a Lie algebra, but the off-diagonal operator A = ddx ⊗ (cid:18) (cid:19) = (cid:18) ddxddx (cid:19) ∈ DO ( S , V ) is not an operator in the Lie algebra C ∞ ( S , M n ( C )) ⋊ V ect ( S ) . One can alwaysembed DO ( S ) = DO ( S , C ) into DO ( S , V ) by identifying A ∈ DO ( S ) with A ⊗ Id C n ∈ DO ( S , V ) . This identification is a morphism of unital algebras anda morphism of Lie algebras. It is a straightforward to check that the similar anti-symmetrization of the product (2) gives a Lie algebra structure on Ψ DO ( S ) andthe algebra DO ( S ) is a Lie subalgebra in it.One of the most exciting properties of this pair of infinite-dimensional Lie al-gebras is an existence of a trace functional (which is quite atypical in the infinite-dimensional world). This functional is known as Adler trace
Tr( A ) = I S tr n ( a − ( x )) dx, where tr n is the classical trace of n × n matrices, and it defines a bilinear invariantsymmetric form on Ψ DO ( S )( A, B ) → Tr( A ◦ B ) , A, B ∈ Ψ DO ( S ) , which is invariant with respect the multiplication: ( C ◦ A, B ) = (
A, B ◦ C ) and alsoinvariant with respect to the Lie bracket: ([ C, A ] , B ) = ( A, [ B, C ]) for any triple A, B, C ∈ Ψ DO ( S ) . This form is a non-degenerate and can be used to build aninjective map from the algebra Ψ DO ( S ) with its dual : to each A ∈ Ψ DO ( S ) onecan assign the linear functional l A ∈ (Ψ DO ( S )) ∗ such that l A ( X ) = Tr( A ◦ X ) Let A ∈ Ψ DO ( S ) such that it contains only negative degrees of the symbol D = ∂ : A = − X k = −∞ b k ( x ) ∂ k . Such "purely Integral"operators are also closed with respect to both operations ◦ and [ , ] and we shall denote this subalgebra in Ψ DO ( S ) by IO ( S ) . It is easy tocheck that the subalgebra DO ( S ) is dual to the subalgebra IO ( S ) via the bilinear EALIZATION OF QUATERNIONIC KP HIERARCHY 11 invariant form ( − , − ) and the "full" algebra Ψ DO ( S ) = DO ( S ) ⊕ IO ( S ) . Bothsubalgebras are isotropic with respect to ( − , − ) . The algebra triple (Ψ DO ( S ) , DO ( S ) , IO ( S )) is known as a Manin triple and the algebra DO ( S ) carries a structure of a Liebialgebra . We should admit that strictly speaking this triple and this bialgebra arenot a genuine example of both structures in view of the following remark:
Remark 1.17.
We should remark that while DO ( S ) = ( IO ( S )) ∗ the natural map IO ( S ) → ( DO ( S )) ∗ is not surjective since not every continuous linear functionalon C ∞ ( S ) is of the form F → ( F, f ) , F ∈ C ∞ ( S ) ( [8] ). In what follows by abuse of the rigorous terminology ("pseudo-Manin triple","pseudo-Lie bialgebra", "Khovanova triple" etc.) we shall call the operator tripleabove by Manin triple and refer DO ( S ) as a Lie bialgebra.1.3.2. Differential and integral part.
We first remind that if V = C n and use thenotations DO ( S , V ) = [ o ∈ N X ≤ k ≤ o a k ∂ k , IO ( S , V ) = X k ≤− a k ∂ k we get also the vector space decomposition Ψ DO ( S , V ) = DO ( S , V ) ⊕ IO ( S , V ) . (3)such that any (matrix) order k pseudo-differential operator A = P ki = −∞ a i ∂ i issplitted in two components A = A + + A − with A + = P ki =0 a i ∂ i and A − = P − i = −∞ a i ∂ i . In that case, when V = C n and with obvious extension of notations,the algebra triple (cid:0) Ψ DO ( S , V ) , DO ( S , V ) , IO ( S , V ) (cid:1) is known as a Manin tripleand the algebra DO ( S , V ) carries a structure of a Lie bialgebra. We shall use also(by abuse of notation) the notation Res( A ) for the residue-matrix function : Res : M n (Ψ DO ( S , C )) → C ∞ ( S , M n ( C )) , , A → a − ( x ) Let
A, B be some matrix-valued pseudo-differential operators, such that A = P ki = −∞ a i ∂ i , B = P lj = −∞ b j ∂ i with a j , b j some matrix-valued functions. Thenit is a straightforward exercise to check that there exists a matrix-valued function F such that Tr([
A, B ]) = I tr n (Res[ A, B ]) = I dF = 0 . Remark 1.18.
The same holds when we replace concrete algebras of functions C ∞ ( S ) by an abstract associative algebra R with unit element, equipped with in-tegration properties, we refer to [28, 29] for a detailed description for the corre-sponding algebra of formal operators Ψ DO ( R ) . Then, in presence of a non-trivialone-form H : R → C , one can define an analogous ot the Adler map that we notealso Tr by Tr : X k ∈ Z a k ∂ k I a − . For example, when R = C ∞ ( S , M n ( C )) for n ≥ , i.e. when Ψ DO ( R ) = Ψ DO ( S , C n ) = M n (Ψ DO ( S , C )) , the natural 1-form H on R is exactly H S ◦ tr n already described. AND VLADIMIR ROUBTSOV , , Poisson structures on matrix pseudo-differential operators. . In anal-ogy with the "scalar" ( n = 1 ) case one can define the first and the second Gelfand-Dikii Poisson structures in the framework of the formal Gelfand "variational"differential-geometric formalism in the infinite-dimensional setting. The resultsof this subsection are not new and are well-known since almost 30 years (see forexample [3]). We define an infinite dimensional affine variety L k whose points,monic differential operators of order k , are defined by k matrix function coefficients ¯ u = ( u ( x ) , . . . , u k ( x )) such that ∀ j : 1 ≤ j ≤ k, u j ( x ) ∈ C ∞ ( S , M n ( C )) : L k = { L = ∂ k + u ∂ k − + . . . + u k } . We consider a function algebra C ( L k ) as a set of functionals l : L k → C of type l [¯ u ] := I tr(pol( ∂ αx ( u j ))) , where pol( ∂ αx ( u j )) is a differential polynomial on u j ( x ) . The tangent space to L k consists of differential operators of order k − and the cotangent space T ∗ L k canbe identified with the quotient IO ( S , V ) /IO − k ( S , V ) : via the coupling T ∗ L k × T L k → C ( L k ) , h X, V i = Tr( X ◦ V ) . Here X ∈ T ∗ L k is the set of "covectors" ofthe type X = P kj =1 ∂ − jx ◦ p j , p j ∈ pol( ∂ αx ( u j ) . We shall remind the definition of variational derivative of a functional l [¯ u ] ∈ C ( L ) : δl [¯ u ] δu j ( x ) pq = ∞ X s = o ( − s d r dx r ∂ tr(pol)(¯u)(x) ∂ ( u ( s ) j ) pq ! , ≤ p, q ≤ n. The variational derivative assigns to each functional l [¯ u ] ∈ C ( L ) the pseudo-differentialoperator X l = k X r =0 ∂ − r (cid:18) δl [¯ u ] δu k +1 − r (cid:19) . Let X l , be two such operators which can be interpreted as two covectors on T ∗ L k .We define a family of brackets {− , −} λ : C ( L ) × C ( L ) → C ( L ) : { l , l } λ ( L ) = I tr n (Res(( L + λ )( X l ( L + λ )) + X l − (( L + λ ) X l ) + ( L + λ ) X l ) = { l , l } ( L ) + λ { l , l } ( L ) = I tr n (Res( L ( X l L ) + X l − ( LX l ) + LX l )) + λ I tr n (Res([ L, X l ] + X l ) . Theorem 1.19. (Adler-Gelfand-Dickey) (1)
The family {− , −} λ is a family of Poisson structures on C ( L ) ; (2) The corresponding Hamiltonian map H λ : T ∗ L k → T L k : is given by H λ ( X ) = ( LX ) + L − L ( XL ) + + λ [ L, X ] + , X ∈ T ∗ L k , L ∈ L k . (3) H λ ( X ) = H ( X )+ λH ( X ) and each V i , i = 1 , are Hamiltonian mappings. (4) The Hamiltonian maps H i relate to the Poisson brackets via { l , l } λ ( L ) = H λ ( δl )( l ) . EALIZATION OF QUATERNIONIC KP HIERARCHY 13 (5)
Covector fields T ∗ L k carry a Lie algebra structure with the bracket [ X, Y ] = [( XL ) + Y + ( Y L ) − X − X ( LY ) − − Y ( LX ) + + H ( X ) Y ) − H ( Y )( X )] − which will be called the second Gelfand-Dikii algebra GD . This structure relates in some sense to the Manin triple on Ψ DO ( S , V ) . Semenov-Tyan-Shansky r − matrix construction. Let A ± two elements of theLie algebra Ψ DO ( S , V ) such that A + ∈ DO ( S , V ) and A − ∈ IO ( S , V ) . Thenone can identify Ψ DO ( S , V ) ⊗ Ψ DO ( S , V ) with Hom(Ψ DO ( S , V ) , Ψ DO ( S , V ) using the inner product on Ψ DO ( S , V ) . Therefore, if we consider the bi-vector r ∈ Λ (Ψ DO ( S , V )) such that h r , A ∗ + ∧ A ∗− i = ( A + , A − ) = Tr( A + ◦ A − ) , where A ∗ is a dual to A with respect to the inner product., then we can identify it withthe operator ˜ r ∈ End(Ψ DO ( S , V )) such that ˜ r | DO ( S ,V ) = 1 , ˜ r | IO ( S ,V ) = − . Analogues of splittings.
Back to F Cl ( S , V ) , the maps A ∈ F Cl ( S , V ) X k ∈ Z σ k ( A )( x, ∂ k and A ∈ F Cl ( S , V ) X k ∈ Z σ k ( A )( x, − ∂ k , identify Ψ DO ( S , V ) with F Cl + ( S , V ) for the first one and F Cl − ( S , V ) for thesecond one.Thus, there exists a decomposition F Cl + ( S , V ) = F Cl + ,D ( S , V ) ⊕F Cl + ,S ( S , V ) and another F Cl − ( S , V ) = F Cl − ,D ( S , V ) ⊕ F Cl − ,S ( S , V ) , and setting F Cl D ( S , V ) = F Cl + ,D ( S , V ) ⊕ F Cl − ,D ( S , V ) , F Cl S ( S , V ) = F Cl + ,S ( S , V ) ⊕ F Cl − ,S ( S , V ) , we get the vector space decomposition analogous to (3): F Cl ( S , V ) = F Cl D ( S , V ) ⊕ F Cl S ( S , V ) , Manin pairs on F Cl ( S , V ) . Extension of the classical Manin triple to F Cl ( S , V ) . The Adler trace [1]defined by
T r : A = X k ≤ o a k ∂ k Z S tr ( a − ) is the only non trivial trace on Ψ DO ( S , V ) . Morover, see e.g. [11] and [19],
Theorem 1.20. (Ψ DO ( S , V ) , IO ( S , V ) , DO ( S , V ) , T r ) is a Manin triple. The Wodzicki residue ([35], see e.g. [15]) is usually known as an “extension”of the Adler trace to F Cl ( S , V ) and hence to Cl ( S , V ) . For the sake of deeperinsight on what is described in the rest of this paper, we need to precise that thespace of traces on F Cl ( S , V ) is 2-dimensional, generated by two functionals: res + : A Z S σ − ( A )( x, | dx | AND VLADIMIR ROUBTSOV , , and res − : A Z S tr ( σ − ( A ))( x, − | dx | . The functionals res ± are the only non-vanishing traces on F Cl ± ( S , V ) (up to ascalar factor) and are vanishing on F Cl ∓ ( S , V ) . The (classical) Wodzicki residuereads as res = res + + res − . Because the partial symbol σ − ( A ) of an opera-tor A ∈ Ψ DO ( S , V ) is skew-symmetric in the ξ − variable, res is vanishing on Ψ DO ( S , V ) = F Cl ee ( S , V ) , so that it is superficial to state that the Wodzickiresidue is “simply” the extension of the Adler trace. However the two linear func-tionals already described, namely A ∈ F Cl ( S , V ) X k ∈ Z σ k ( A )( x, ∂ k and A ∈ F Cl ( S , V ) X k ∈ Z σ k ( A )( x, − ∂ k , identity res + and res − respectively with T r.
By the way, we can state:
Theorem 1.21.
We have three Manin triples: ( F Cl + ( S , V ) , F Cl + ,S ( S , V ) , F Cl + ,D ( S , V ) , res + ) , ( F Cl − ( S , V ) , F Cl − ,S ( S , V ) , F Cl − ,D ( S , V ) , res − ) and ( F Cl ( S , V ) , F Cl S ( S , V ) , F Cl D ( S , V ) , res ) . A remark on two "non-invariant Manin triples".
Following [11], given an op-erator r acting on F Cl ( S , V ) satisfying r = Id, one can form a F Cl ( S , V ) − valuedskew-symmetric bilinear form [ ., . ] r = 12 ([ r ( . ) , . ] + [ ., r ( . )]) . In what follows, we concentrate on the cases r = ǫ ( D ) ◦ ( . ) , r = s and also r = s ′ . The corresponding brackets will be noted respectively by [ ., . ] ǫ ( D ) , [ ., . ] s and [ ., . ] s ′ . Let us define ( A, B ) s ′ = res ( A, s ′ ( B )) . By direct calculations, we find succes-sively:
Lemma 1.22. ( . ; . ) s ′ is non degenerate and symmetric. Theorem 1.23. On F Cl ( S , V ) = F Cl + ( S , V ) + F Cl − ( S , V ) , ( . ; . ) s ′ is a nondegenerate and symmetric bilinear from for which the Lie algebras F Cl + ( S , V ) and F Cl − ( S , V ) are isotropic. Let us define ( A, B ) s = res ( A, s ( B )) . Lemma 1.24. ( . ; . ) s is non degenerate and skew-symmetric but neither invariantfor [ ., . ] nor for [ ., . ] ǫ ( D ) . Theorem 1.25. On F Cl ( S , V ) = F Cl + ( S , V )+ F Cl − ( S , V ) , ( . ; . ) s is a non de-generate and skew-symmetric bilinear from for which the Lie algebras F Cl + ( S , V ) and F Cl − ( S , V ) are isotropic. EALIZATION OF QUATERNIONIC KP HIERARCHY 15
Two other Manin pairs.
Let us consider the decomposition F Cl ( S , V ) = F Cl ee ( S , V ) + F Cl eo ( S , V ) , that we equip with the classical Lie bracket [ ., . ] or with [ ., . ] ǫ ( D ) . and with thebilinear form ( A, B ) = res ( AB ) . Theorem 1.26. res(AB) is a bilinear, non degenerate, symmetric and invariantform for both brackets, and F Cl ee ( S , V ) as well as F Cl eo ( S , V ) are isotropicvector spaces. Moreover, • for [ ., . ] , F Cl ee ( S , V ) is a Lie algebra • for [ ., . ] ǫ ( D ) , F Cl eo ( S , V ) is a Lie algebra. Polarized Lie bracket.
The modified Yang–Baxter equation gives the con-dition on r for making [ ., . ] r a Lie bracket: [ r X, r Y ] − r ([ r X, Y ] + [ X, r Y ]) = − [ X, Y ] . By direct computations, we get the following:
Theorem 1.27.
On the vector space F Cl ( S , V ) , (1) [ ., . ] ǫ ( D ) is a Lie bracket for which [ F Cl ee ( S , V ) , F Cl ee ( S , V )] ǫ ( D ) ⊂ F Cl eo ( S , V ) and [ F Cl eo ( S , V ) , F Cl eo ( S , V )] ǫ ( D ) ⊂ F Cl eo ( S , V ) . (2) [ ., . ] s and [ ., . ] s ′ are not Lie brackets. Remark 1.28. (Testing Rota-Baxter equations and Reynolds operators)
Testing by direct calculations the Rota-Baxter equations R ( u ) R ( v ) − R ( R ( u ) v ) − R ( uR ( v )) = λR ( uv ) for a weight λ ∈ C , one finds that R = ǫ ( D ) ◦ ( . ) , R = s and R = s ′ do not satisfythe Rota-Baxter equations (i.e. don’t define new associative algebra operations)The same calculations show that these are not Reynolds operators (i.e. they do notsatisfy the condition R ( R ( u ) v ) = R ( u ) R ( v ) for all u, v in the underlying associativealgebra). Preliminaries on the KP hierarchy.
Let R be an algebra of functionsequipped with a derivation ∂. For us, R = C ∞ ( S , K ) with K = R , C and H , and ∂ = ddx . In this context, where algebras of functions R are Fréchet algebras, a naturalnotion of differentiability occurs, making addition, multiplication and differentiationsmooth. By the way, considering addition and multiplication in Ψ DO ( S , K ) , onecan say that addition and multiplication in Ψ DO ( S , K ) by understanding, underthis terminology, that, if A = P n ∈ Z a n ∂ n and B = P n ∈ Z b n ∂ n , setting A + B = C = P n ∈ Z c n ∂ n and AB = D = P n ∈ Z d n ∂ n the map (( a n ) n ∈ Z , ( b n ) n ∈ Z ) (( c n ) n ∈ Z , ( d n ) n ∈ Z ) is smooth in the relevant infinite product. We make these precisons in other tocircumvent the technical tools recently developed in [10, 25] where a fully rigorousframework for smoothness on these objects is described and used. Let T = { t n } n ∈ N ∗ be an infinite set of formal (time) variables and let us consider the algebra of formalseries Ψ DO ( S , K )[[ T ]] with infinite set of formal variables t , t , · with T − valuation val defined by val T ( t n ) = n [28]. One can extend naturally on Ψ DO ( S , K )[[ T ]] AND VLADIMIR ROUBTSOV , , the notion of smoothness from the same notion on Ψ DO ( S , K ) , see [25] for a morecomplete description. The Kadomtsev-Petviashvili (KP) hierarchy reads(4) dLdt k = (cid:2) ( L k ) D , L (cid:3) , k ≥ , with initial condition L (0) = L ∈ ∂ + Ψ − ( R ) . The dependent variable L is chosento be of the form L = ∂ + P α ≤− u α ∂ α ∈ Ψ ( S , K )[[ T ]] . A standard referenceon (4) is L.A. Dickey’s treatise [6], see also [19, 28, 29]. In order to solve the KPhierarchy, we need the following groups (see e.g. [25] for a latest adaptation ofMulase’s construction [28, 29]): ¯ G = 1 + Ψ DO − ( S , K )[[ T ]] , Ψ = ( P = X α ∈ Z a α ∂ α ∈ Ψ( S , K )[[ T ]] : val T ( a α ) ≥ α and P | t =0 ∈ DO − ( S , K ) ) and D = ( P = X α ∈ Z a α ∂ α : P ∈ Ψ( A t ) and a α = 0 for α < ) . We have a matched pair
Ψ = ¯
G ⊲⊳ D which is smooth under the terminology we gavebefore. The following result, from [25], gives a synthesied statement of main resultson the KP hierarchy (4) and states smooth dependence on the initial conditions inthe case where R is commutative (i.e. R = C ∞ ( S , R ) or R = C ∞ ( S , C ) in thiswork). Theorem 1.29. [25]
Consider the KP hierarchy 4 with initial condition L (0) = L .Then, (1) There exists a pair ( S, Y ) ∈ ¯ G ×D such that the unique solution to Equation (4) with L | t =0 = L is L ( t , t , · · · ) = Y L Y − = SL S − . (2) The pair ( S, Y ) is uniquely determined by the smooth decomposition problem exp X k ∈ N τ k L k ! = S − Y and the solution L depends smoothly on the initial condition L . (3) The solution operator L is smoothly dependent on the initial value L . We now describe the case K = H = R + i R + j R + k R . The algebra Ψ DO ( S , H ) is constructed from the non commutative Fréchet algebra C ∞ ( S , H ) = C ∞ ( S , R ) ⊕ iC ∞ ( S , R ) ⊕ jC ∞ ( S , R ) ⊕ kC ∞ ( S , R ) . All the constructions before remain valid following [20, 14], setting V = H as a4-dimensional R − algebra, and the algebraic description of the solutions of the KPhierarchy (4) with L ∈ Ψ DO ( S , H ) and L ∈ Ψ DO ( S , H )[[ T ]] as before can becompleted by stating that the coefficients of the T − series of L depend smoothly onthe initial value L from [10]. EALIZATION OF QUATERNIONIC KP HIERARCHY 17 Injecting Ψ DO into F Cl.
Injecting Ψ DO ( S , K ) in F Cl ( S , K ) . . We already mentionned the identifi-cation of Ψ DO ( S , K ) with F Cl ee ( S , K ) , present when K = R or C in [26]. Weclaim here that this identification also applies straightway when K = H . We denoteby Φ ee this identification, that can be generalized to Φ ee,λ : X k ∈ Z a k (cid:18) ddx (cid:19) k ∈ Ψ DO ( S , K ) X k ∈ Z a k (cid:18) λ ddx (cid:19) k ∈ F Cl ee ( S , K ) . Similar to this identification, we have other injections for λ ∈ R ∗ :Φ ǫ ( D ) ,λ : X k ∈ Z a k (cid:18) ddx (cid:19) k ∈ Ψ DO ( S , K ) X k ∈ Z a k (cid:18) λǫ ( D ) ddx (cid:19) k ∈ F Cl ( S , K ) , and Φ λ,µ : X k ∈ Z a k (cid:18) ddx (cid:19) k ∈ Ψ DO ( S , K ) X k ∈ Z a k λ k (cid:18) ddx (cid:19) k + + µ k (cid:18) ddx (cid:19) k − ! ∈ F Cl ( S , K ) for ( λ, µ ) ∈ C \{ (0; 0) } , with unusual convention k = 0 ∀ k ∈ Z . Remark 2.1. Φ , = Φ ee and Φ , − = Φ ǫ ( D ) , . Remark 2.2. Im Φ , = F Cl + ( S , K ) and Φ , is a isomorphism of algebrasfrom Ψ DO ( S , K ) to F Cl + ( S , K ) . The same way, Φ , identifies the algebras Ψ DO ( S , K ) and F Cl − ( S , K ) . Remark 2.3.
Wa have also to say that the maps Φ λ,µ are not algebra morphismsunless ( λ, µ ) ∈ { (1; 0) , (0; 1) , (1; 1) } . For example, let λ ∈ C − {
0; 1 } . the map Φ λ, pushes forward the multiplication on Ψ DO ( S , K ) to a deformed composition ∗ k on F Cl + ( S , K ) that reads as σ ( A ) ∗ k σ ( B ) = P α ∈ N ( − i ) α α ! .k α D αx σ ( A ) D αξ σ ( B ) . Let us now give some sample images: A ∈ Ψ DO ( S , C ) 1 ddx − ddx = ∆ (cid:0) ddx (cid:1) − Φ ǫ ( D ) , ( A ) 1 ǫ ( D ) ddx = i | D | ∆ (cid:0) ǫ ( D ) ddx (cid:1) − Φ ee, − ( A ) 1 − ddx ∆ (cid:0) − ddx (cid:1) − Φ , ( A ) 1 + (cid:0) ddx (cid:1) + ∆ + (cid:16)(cid:0) ddx (cid:1) − (cid:17) + From our previous remarks, we get:
Theorem 2.4.
The map Φ , × Φ , : Ψ DO ( S , K ) → F Cl + ( S , K ) × F Cl − ( S , K ) = F Cl ( S , K ) is an isomorphism of algebra. AND VLADIMIR ROUBTSOV , , We also remark a new subalgebra of F Cl ( S , K ) : Definition 2.5.
Let F Cl ǫ ( S , K ) be the image of Φ ǫ ( D ) , in F Cl ( S , K ) . We have the obvious identification F Cl ǫ ( S , K ) = C ∞ ( S , K )(( i | D | − )) as avector space.2.2. Identification of F Cl ( S , C ) with Ψ DO ( S , H ) . . Let iǫ ( D ) = (cid:18) ddx (cid:19) . | D | − = | D | − . (cid:18) ddx (cid:19) . We define the operator J = iǫ ( D ) ◦ ( . ) on F Cl ( S , V ) . Theorem 2.6.
The operator J defines an integrable almost complex structure on F Cl ( S , V ) . F Cl ( S , V ) = Ψ DO ( S , V ) ⊗ C as a real algebra, identifying F Cl e e ( S , V ) with Ψ DO ( S , V ) (real part) and F Cl e o ( S , V ) with i Ψ DO ( S , V ) (imaginary part). We now identify two other almost complex structures: J = is ( . ) , J = is ′ andClearly, ∀ i ∈ {
2; 3 } , J i = − Id and we have also: Proposition 2.7. J ◦ J = − J ◦ J Theorem 2.8.
The operator J defines a non integrable almost complex structureon F Cl ( S , V ) . Hence, gathering all these results, we get that the almost quater-nionic structure ( J , J ) is non integrable. Proposition 2.9. J ◦ J = − J ◦ J Proposition 2.10. J ◦ J = J J = − J ◦ J Theorem 2.11.
The operator J defines a non integrable almost complex structureon F Cl ( S , V ) . The almost quaternionic structure ( J , J ) is non integrable. Let us now define J = J J . Proposition 2.12.
We have: • J = − Id. • J J = − J J . • J J = − J J . KP hierarchy with integer and complex order Lax operators in F Cl ( S , C ) and Ψ DO ( S , H ) . Multiple classical KP hierarchies on F Cl ( S , K ) . The (classical) KP hi-erarchy on Ψ DO ( S , K ) can then push-forward on F Cl -classes of operators byvarious ways: • via identifications of Ψ DO ( S , K ) with subalgebras or ideals of F Cl ( S , K ) , for K = R , C or H . • by changing the standard multiplication of F Cl ( S , K ) for K = R , C or H , by “twisting it” by the operator ǫ ( D ) or iǫ ( D ) . • via the almost quaternionic structures that we identified on F Cl ( S , C ) inorder to identify it with Ψ DO ( S , H ) Let us describe in a detailed way these different approaches.
EALIZATION OF QUATERNIONIC KP HIERARCHY 19
Push-Forward via Φ λ,µ maps. Let K = C or H . For each choice of ( λ, µ ) ∈ C \{
0; 0 } identifies ddx ∈ Ψ DO ( S , K ) with an operator in F Cl ( S , K ) with thesame algebraic properties. Notation: ∂ λ,µ = Φ λ,µ (cid:0) ddx (cid:1) and F Cl λ,µ ( S , K ) = Im Φ λ,µ . Then we can develop the KP hierarchy on F Cl λ,µ ( S , K ) . We first remark that,since each map Φ λ,µ is a degree morphism of filtered algebras, each push-forwardof the unique solotion L of the KP hierachy (4) generates a solution of the corre-sponding equation in F Cl ( S , K ) which reads the same way: dLdt k = (cid:2) ( L k ) D , L (cid:3) , k ≥ , where solutions operators now belong to F Cl ( S , K )[[ T ]] and where each initialvalue Φ λ,µ ( L ) ∈ ∂ λ,µ + F Cl − λ,µ ( S , K ) with obvious extension of notations. There-fore, for any initial value L ∈ Ψ DO ( S , K ) , we get a family of operators L λ,µ ∈ F Cl λ,µ ( S , K )[[ T ]] ⊂ F Cl ( S , K )[[ T ]] parametrized by the complex parameters λ and µ chosen as before, which satisfiesthe KP hierarchy in F Cl ( S , K ) and with initial values Φ λ,µ ( L ) . Existence, uniqueness and well-posedness of the KP system in F Cl ( S , K ) . . We adapt here the r − matrix approach for the construction of the solutions, alongthe lines of [10] with the following specific choices: • The algebra of smooth coefficients for formal pseudo-differential operatorsis R = C ∞ ( S , M n ( K )) ⊕ ǫ ( D ) C ∞ ( S , M n ( K )) with multiplication rulesinherited from Cl ( S , K n ) . • The differential operator is ∂ = ddx . Proposition 3.1. Ψ DO ( R ) = F Cl ( S , K n ) and there is an identification of theManin triples (Ψ DO ( R ) , DO ( R ) , IO ( R )) with ( F Cl ( S , K n ) , F Cl D ( S , K n ) , F Cl S ( S , K n )) . Hence, applying the main result of [31] completed, for well-posedness, by [10,Theorem 4.1] or by [25, Theorem 4.1] when R = C ∞ ( S , K ) = M ( C ∞ ( S , K )) isa commutative algebra, we can state the following: Proposition 3.2.
The Kadomtsev-Petviashvili (KP) hierarchy (4) on Ψ DO ( R ) (resp. F Cl ( S , K n ) ) with initial condition L (0) = L ∈ ∂ + Ψ DO − ( R ) (resp. ∈ ∂ + F Cl − ( S , K n ) ) satisfies Theorem 1.29. Remark 3.3.
We have used here, intrinsically, the integrable almost complex struc-ture J . Indeed, R = C ∞ ( S , M n ( K )) + J C ∞ ( S , M n ( K )) is an algebra. Remark 3.4.
There exists another way to justify Proposition 3.2. One can usealternatively the splitting F Cl ( S , K n ) = F Cl + ( S , K n ) ⊕ F Cl − ( S , K n ) . Then Equation (4) on F Cl ( S , K n ) splits into two independent equations, similarto Equation (4) on F Cl ± ( S , K n ) . Through the identification maps Φ , and Φ , of F Cl ± ( S , K n ) with Ψ DO ( S , K n ) , we get existence, uniqueness and well-posednessfor Equation (4) on F Cl ( S , K n ) with initaial value L ∈ ∂ + F Cl − ( S , K n ) . From this last remark, we can generalize the identification procedure, changingthe maps Φ ee , Φ , and Φ , by the family of maps Φ λ,µ . AND VLADIMIR ROUBTSOV , , Theorem 3.5.
Let ( λ, µ ) ∈ ( C ∗ ) . Then the KP equation (4) in F Cl ( S , K n ) with initial value L ∈ ∂ λ,µ + F Cl − ( S , K n ) has an unique solution L in ∂ λ,µ + F Cl − ( S , K n )[[ T ]] and the problem is well-posed: the solution L depends smoothlyon L . Twisted KP hierarchy.
Let us now change the standard multiplication on F Cl ( S , K ) by ( A, B ) ǫAB where ǫ = ǫ ( D ) or aǫ ( D ) for any a ∈ C ∗ . Since ǫ ( D ) commuteswith any element of F Cl ( S , K ) for the standard multiplication, this new multipli-cation defines a new algebra structure on F Cl ( S , K ) . When necessary we note by ◦ the standard multiplication, and by ◦ ǫ the twisted one. Associated to this multi-plication, we get the deformed Lie bracket [ ., . ] ǫ . Then we get again and equationsimilar to (4)(5) dLdt k = (cid:2) ǫ k − ( L k ) D , L (cid:3) ǫ = ǫ k (cid:2) ( L k ) D , L (cid:3) , k ≥ , where powers in this equation are taken with respect to ◦ . Theorem 3.6.
The Let L such that L ∈ ∂ λ,µ + F Cl − ( S , K n ) , with ( λ, µ ) ∈ ( C ∗ ) . Then the ǫ − KP hierarchy (5) with initial value L has an unique solution.Moreover, the problem is well-posed. KP hierarchies with complex powers.
We finally extend all the construc-tions of the last section to complex powers, along the lines of [11]. Let K = C or H . We consider an operator L of complex order α such that(6) L ∈ (cid:18) ddx (cid:19) α + F Cl α − ( S , K ) or(7) L ∈ | D | α + F Cl α − ( S , K ) For each setting (6) and (7), we define the complex KP hierarchy on F Cl α ( S , K ) by(8) dLdt k = h ( L k/α ) D , L i ǫ = − (cid:2) ( L k ) S , L (cid:3) , k ≥ , where L k/α = exp (cid:0) kα log L (cid:1) and the solution L ∈ F Cl α ( S , K )[[ T ]] . Theorem 3.7.
The KP hierarchy (8) with initial value L defined along the linesof (6) or (7) has an unique solution in F Cl α ( S , K )[[ T ]] . Moreover, the prblem iswell-posed. Hamiltonian approaches
Now we consider F Cl ( S , K n ) and we define the regular dual space F Cl ( S , K n ) ′ = { µ ∈ L ( F Cl ( S , K n ) , K ) : µ = h P, ·i for some P ∈ F Cl ( S , K n ) } . We can adapt standard results described in section 1.4 of Hamiltonian mechanicsas follows: let f : F Cl ( S , K n ) ′ → B be a polynomial function of the type f ( µ ) = n X k =0 a k res + ( P k ) + n X k =0 b k res − ( P k ) = res n X k =0 a k P k + + b k P k − ! with µ = h P, . i . In our picture, the decomposition F Cl ( S , K n ) = F Cl + ( S , K n ) ⊕F Cl − ( S , K n ) that we use extensively all along this work carry a residue trace on EALIZATION OF QUATERNIONIC KP HIERARCHY 21 each component of the decomposition. These are these two residues, res + and res − , that replace Res in the constructions of section 1.4. Under these assumptions, we de-fine the same way the functional derivative and the pairing < . | . > of F Cl ( S , K n ) ′ with F Cl ( S , K n ) . The decomposition F Cl ( S , K n ) = F Cl D ( S , K n ) ⊕F Cl S ( S , K n ) allows us to consider a new Lie bracket on the regular dual space F Cl ( S , K n ) ′ givenby [ P, Q ] = [ P D , Q D ] − [ P S , Q S ] , This bracket determines a new Poisson structure { , } on F Cl ( S , K n ) ′ , simply by replacing the original Lie product for [ , ] . Usingagain the non-degenerate pairing we get: Lemma 4.1.
Let H : F Cl ( S , K n ) ′ → K be a smooth function on F Cl ( S , K n ) ′ such that (9) (cid:28) µ (cid:12)(cid:12)(cid:12)(cid:12) (cid:20) δHδµ , · (cid:21)(cid:29) = 0 for all µ ∈ F Cl ( S , K n ) ′ . Then, as equations on F Cl ( S , K n ) , the Hamiltonian equations of motion withrespect to the { , } Poisson structure of F Cl ( S , K n ) ′′ are (10) d Pd t = "(cid:18) δHδµ (cid:19) + , P . We now use some specific functions H . Let us recall the following results (seefor example [6] or the more recent review [9]): Proposition 4.2.
We define the functions H k ( L ) = T race (cid:0) ( L k ) (cid:1) , k = 1 , , , · · · , for L ∈ F Cl ( S , K n ) . Then, δH k δL = kL k − . In particular, the functions H k satisfy (9) . Thus, we can apply Lemma 4.1. It yields:
Proposition 4.3.
Let us equip the Lie algebra F Cl ( S , K n ) with the non-degeneratepairing ( a, b ) res ( ab ) . Write F Cl ( S , K n ) = F Cl D ( S , K n ) ⊕ F Cl S ( S , K n ) andconsider the Hamiltonian functions (11) H k ( µ ) = 1 k res W (cid:0) ( L k +1 ) (cid:1) for µ = h L, . i . The corresponding Hamiltonian equations of motion with respect tothe { , } Poisson structure of F Cl ( S , K n ) ′ are dLdt k = (cid:2) ( L k ) D , L (cid:3) . Following now [11] we get the Hamiltonian formulation of the KP hierarchy withcomplex powers: For this, we need to generalize the Gelfand-Dickii stricture eitherto L α = (cid:18) ddx (cid:19) α + F Cl α − ( S , K n ) or to L ′ α = (cid:12)(cid:12)(cid:12)(cid:12) ddx (cid:12)(cid:12)(cid:12)(cid:12) α + F Cl α − ( S , K n ) . In both case, we specialize our computations to F Cl α + ( S , K n ) , and with F Cl α − ( S , K n ) respectively, which both identify with Ψ DO α ( S , K n ) . Under these identifications,the computations described in [11, pp 55–57]:
Theorem 4.4. On L α and on L ′ α , the Hzmiltonian vector field associated to H k = αk resL k/α is V = h L k/αD , L i . AND VLADIMIR ROUBTSOV , , Appendix: Proofs
We collect in the Appendix all routine and technical prooofs which are oftennothing but a straightforward verification. The end of the Appendix contains alsotechnical proofs of some theorems about KP hierarhies which are very similar ide-ologically of our ancient proofs from [11].5.1.
Proofs of section 1.
Lemma 1.22.
Since res is non degenerate then ( . ; . ) s ′ is non degenerate. Moreover,identifying F Cl + ( S , V ) and F Cl − ( S , V ) as two copies of Ψ DO ( S , V ) , writingby T r the Adler trace on the latter one, we have that res ( A, s ′ ( B )) = T r ( A + B − )+ T r ( A − B + ) = T r ( B + A − )+ T r ( B − A + ) = res ( B, s ′ ( A )) which proves symmetry. (cid:3) Lemma 1.24.
Since res is non degenerate then ( . ; . ) s is non degenerate. Moreover, res ( A, s ( B )) = res (( A ee + A eo ) + ( B ee − B eo ))= res ( A eo B ee ) − res ( A ee B eo )= − res ( B, s ( A )) which proves skewsymmetry. res ( A, s ([ B, C ])) = res (( A ee + A eo ) + ([ B, C ] ee − [ B, C ] eo ))= res ( A eo B ee C ee ) + res ( A eo B eo C eo ) − res ( A ee B ee C eo ) − res ( A ee B eo C ee ) − res ( A eo C ee B ee ) − res ( A eo C eo B eo ) + res ( A ee C ee B eo ]) + res ( A ee C eo B ee ]) while, with the same calculations, res ([ A, C ] , s ( B )) = res ( A eo C ee B ee ) + res ( A ee C eo B ee ) − res ( A ee C ee B eo ) − res ( A eo C eo B eo ) − res ( A eo B ee C ee ) − res ( A ee B ee C eo ) + res ( A ee B eo C ee ) + res ( A eo B eo C eo ) Let us investigate the same properties with [ ., . ] ǫ ( D ) : res ( A, s ([ B, C ] ǫ ( D ) )) = res (( A ee + A eo )( − ǫ ( D )[ B, C ] ee + ǫ ( D )[ B, C ] eo ))= res ( ǫ ( D ) A eo B ee C eo ) + res ( ǫ ( D ) A eo B eo C ee ) − res ( ǫ ( D ) A ee B eo C eo ) − res ( ǫ ( D ) A ee B ee C ee ) − res ( ǫ ( D ) A eo C eo B ee ) − res ( ǫ ( D ) A eo C ee B eo )+ res ( ǫ ( D ) A ee C eo B eo ) + res ( ǫ ( D ) A ee C ee B ee ) while res ([ A, C ] , s ( B )) = res ( ǫ ( D ) A ee C ee B ee ) + res ( ǫ ( D ) A eo C eo B ee ) − res ( ǫ ( D ) A eo C ee B eo ) − res ( ǫ ( D ) A ee C eo B eo ) − res ( ǫ ( D ) A ee B ee C ee ) − res ( ǫ ( D ) A eo B ee C eo )+ res ( ǫ ( D ) A eo B eo C ee ) + res ( ǫ ( D ) A ee B eo C eo ) (cid:3) Theorem 1.26.
First, F Cl ee ( S , V ) is itself a subalgebra of F Cl ( S , V ) hence ( F Cl ee ( S , V ) , [ ., . ]) is a Lie subalgebra of F Cl ee ( S , V ) on which res ( A, B ) satisfies the same well-known properties: bilinear and symmetric. Moreover, it is well-known that res is EALIZATION OF QUATERNIONIC KP HIERARCHY 23 non-degenerate on F Cl ( S , V ) . From [16], one can deduce by considering only for-mal operators that F Cl ee ( S , V ) is isotropic for res. Secondly, since ǫ ( D ) commuteswith any element of F Cl ( S , V ) , we have that ∀ A, B, C ∈ F Cl eo ( S , V ) , [ A, [ B, C ] ǫ ( D ) ] ǫ ( D ) + [ C, [ A, B ] ǫ ( D ) ] ǫ ( D ) + [ B, [ C, A ] ǫ ( D ) ] ǫ ( D ) = ǫ ( D ) ([ A, [ B, C ]] + [ C, [ A, B ]] + [ B, [ C, A ]]) = 0 , which proves that ( F Cl eo ( S , V ) , [ ., . ] ǫ ( D )) is a Lie bracket. Moreover, A ǫ ( D ) A is a vector space isomorphism from F Cl eo ( S , V ) to F Cl ee ( S , V ) , which implies,with ǫ ( D ) = 1 , that ∀ ( A, B ) ∈ F Cl eo ( S , V ) , res ( AB ) = res (( ǫ ( D ) A )( ǫ ( D ) B )) and shows that F Cl eo ( S , V ) is isotropic for res ( AB ) . Let us finish the proofwith invariance on res.
Invariance with respect to [ ., . ] in F Cl ( S , V ) is well-known since res is tracial. Again since ǫ ( D ) commutates, we have that ∀ ( A, B ) ∈F Cl ( S , V ) , [ A, B ] ǫ ( D ) = [ ǫ ( D ) A, B ] = [
A, ǫ ( D ) B ] hence for ( A, B, C ) ∈ F Cl ( S , V ) ,res ([ A, B ] ǫ ( D ) C ) = res ([ ǫ ( D ) A, B ] C )= res ( B [ C, ǫ ( D ) A ])= − res ( B [ A, C ] ǫ ( D ) ) . (cid:3) Proofs of section 2.2.
Proof of Theorem 2.6.
The operator iǫ ( D ) commutes with any operator u ∈F Cl ( S , V ) . By the way, we simplify the relation that can be found e.g. in [27] thefollowing way: [ u, J v ] + [ J u, v ] = 2 J [ u, v ] and J [ u, v ] − J [ J u, J v ] = J [ u, v ] − J [ u, v ] = 2 J [ u, v ] Hence [ u, J ( v )] + [ J ( u ) , v ] = J [ u, v ] − J [ J ( u ) , J ( v )] which proves integrability. Lemma 5.1.
We have J ( F Cl ee ( S , V )) = F Cl eo ( S , V ) and J ( F Cl eo ( S , V )) = F Cl ee ( S , V ) Proof.
Since J = iǫ ( D ) ◦ ( . ) it follows from the composition rules between even-evenand even-odd class already described. (cid:3) Identifying F Cl eo ( S , V ) with ǫ ( D ) F Cl ee ( S , V ) , we recall that F Cl ( S , V ) = F Cl ee ( S , V ) ⊕ ǫ ( D ) F Cl ee ( S , V ) , we get the complexification result. AND VLADIMIR ROUBTSOV , , Other proofs.Proposition 2.7.
By straightforward computations, we check first that s ( ǫ ( D )) = − ǫ ( D ) . Then, since composition of symbols by ǫ ( D ) is only pointwise multiplication,we get, for a ∈ F Cl ( S , V ) ,J ◦ J ( a )( x, ξ ) = − s ( ǫ ( D ) ◦ a )( x, ξ )= − ǫ ( D )( x, − ξ ) X k ∈ Z ( − k a k ( x, − ξ ) ! (pointwise multiplication) = ǫ ( D )( x, ξ ) s ( a )( x, ξ ) (pointwise multiplication) = − iǫ ( D ) ◦ ( is ( a ))( x, ξ )= − J ◦ J ( a )( x, ξ ) (cid:3) Theorem 2.8. [ u, J v ] + [ J u, v ] = i [ u, v ee ] − i [ u, v eo ] + i [ u ee , v ] − i [ u eo , v ]= i ([ u ee , v ee ] + [ u eo , v ee ] − [ u ee , v eo ] − [ u eo , v eo ]+[ u ee , v ee ] + [ u ee , v eo ] − [ u eo , v ee ] − [ u eo , v eo ])= 2 i ([ u ee , v ee ] − [ u eo , v eo ]) and J [ u, v ] − J [ J u, J v ] = i ([ u ee , v ee ] + [ u eo , v eo ]) − i ([ u ee , v eo ] + [ u eo , v ee ])+ i ([ u ee , v ee ] + [ − u eo , − v eo ]) − i ([ u ee , − v eo ] + [ − u eo , v ee ])= 2 i ([ u ee , v ee ] + [ u eo , v eo ]) As a counter-example, let X = f ( x ) ∂ and let Y = g ( x ) ∂ be two vector fields over S such that [ X, Y ] = 0 . Let u = ǫ ( D ) X ∈ F Cl eo ( S , R ) and let v = ǫ ( D ) Y ∈F Cl eo ( S , R ) . Then J [ u, v ] − J [ J u, J v ] = 2 i [ X, Y ] while [ u, J v ] + [ J u, v ] = − i [ X, Y ] . (cid:3) Proposition 2.9.
By straightforward computations, we check first that s ′ ( ǫ ( D )) = − ǫ ( D ) . Then, since s ′ is a morphism of algebra, J ◦ J ( a ) = − s ′ ( ǫ ( D ) ◦ ( a + , a − )) = − ( − a − , a + )= − ǫ ( D ) ◦ ( − a − , − a + ) = ǫ ( D ) ◦ s ′ ( a + , a − ) = − J ◦ J ( a ) (cid:3) Proposition 2.10.
By straightforward computations, we check first that ss ′ = s ′ s. Then, J J = J J . (cid:3) Theorem 2.11. [ u, J v ]+[ J u, v ] = i ([ u + v − ] + [ u − v + ] , [ u + v − ] + [ u − v + ]) and J [ u, v ] − J [ J u, J v ] = i ([ u + , v + ] + [ u − , v − ] , [ u + , v + ] + [ u − , v − ]) As a counter-example, let X = f ( x ) ∂ and let Y = g ( x ) ∂ be two vector fields over S such that [ X, Y ] = 0 . Let u = X + ∈ F Cl + ( S , R ) and let v = Y + ∈ F Cl + ( S , R ) . Then J [ u, v ] − J [ J u, J v ] = i [ X, Y ] + while [ u, J v ] + [ J u, v ] = 0 . (cid:3) Proposition 2.12. • J = J J J J = − J J J J = − Id • J J = J J J = − J J J = − J J J = − J J EALIZATION OF QUATERNIONIC KP HIERARCHY 25 • J J = J J J = − J J J = − J J (cid:3) Proofs of section 3.
Proof of Proposition 3.1.
From F Cl ( S , K n ) = F Cl ee ( S , K n ) ⊕F Cl eo ( S , K n ) = F Cl ee ( S , K n ) ⊕ iǫ ( D ) F Cl ee ( S , K n ) we get, for A = A ee + A eo ∈ F Cl ee ( S , K n ) ⊕ F Cl eo ( S , K n ) , and for k ∈ Z ,σ k A = σ k ( A ee ) + σ k ( A eo ) = a k,ee ddx k + a k,eo iǫ ( D ) ddx = ( a k,ee + iǫ ( D ) a k,eo ) ∂ k (where ( a k,ee , a k,eo ) ∈ C ∞ ( S , M n ( K )) )which ends the identification of F Cl ( S , K n ) with Ψ DO ( R ) . Since the order of partial symbols is conserved, we get the sameidentifications between F Cl D ( S , K n ) and DO ( R ) , and between F Cl S ( S , K n ) and IO ( R ) . (cid:3) Proof of Theorem 3.5.
We analyze separately the equation on F Cl + ( S , K ) andon F Cl − ( S , K ) . Let us work on F Cl + ( S , K ) . The map Φ , pulls-back the KPhierarchy on Ψ DO ( S , K ) with initial value L ∈ λ∂ + Ψ DO ( S , K ) . When λ = 1 , the classical integration of the KP hierarchy is not achieved by the classical method.However, we use here the scaling first defined to our knowledge in [23]. Let q = λ − . Let ˜ L = qL . Then ˜ L ∈ λ∂ + Ψ DO ( S , K ) , and there exists a Sato operator S ∈ DO − ( S , K ) such that ˜ L = S ∂S − and the KP-system with initialvalue ˜ L has a unique solution ˜ L. We define a λ − scaling in time: t k λ k t k . Following [23], ˜ L ( t , t , ... ) is solution of (4 ) ⇔ L ( t , t , .. ) = λ ˜ L ( λt , λ t , ... ) is solution of (4 ) . The initial value of the solution L is L , which proves existence, uniqueness andsmooth dependence of L on L . Then, we can push-forward the solution L of (4)on Ψ DO ( S , K ) to the solution L + = Φ , ( L ) on F Cl + ( S , K ) . The same procedureholds to get the solution L − on F Cl + ( S , K ) , replacing the constant λ by the con-stant µ. The operator L + + L − furnishes the desired solution of (4) on F Cl ( S , K ) , which is unique and smoothly dependent on the initial value by construction. (cid:3) Proof of Theorem 3.6.
Let us transform slightly Equation 5 for k ∈ N ∗ : dLdt k = ǫ k (cid:2) ( L k ) D , L (cid:3) ⇔ ǫ dLdt k = ǫ k +1 (cid:2) ( L k ) D , L (cid:3) ⇔ d ( ǫL ) dt k = (cid:2) (( ǫL ) k ) D , ( ǫL ) (cid:3) By the way,the field of operators ǫL, with initial value ǫL ∈ ∂ aλ, − aµ + F Cl − ( S , K n ) , is the unique solution of the KP hierarchy (4). Moreover, the map L ǫL issmooth, biunivoque, with smooth inverse, which ends the proof. (cid:3) Proof of Theorem 3.7.
Let us first analyze case (6). Then L /α = exp (cid:18) α log L (cid:19) ∈ ddx + F Cl ( S , K ) . We can then adapt [31] and define the dressing operator U = exp X k ∈ N ∗ t k ( L /α ) k ! AND VLADIMIR ROUBTSOV , , that we decompose into U = U + + U − where U ± = exp X k ∈ N ∗ t k ( L /α ) k ± ! . Working independently on the two components F Cl ± ( S , K ) , Mulase factorizationholds, U ± = S − ± Y ± and setting L ± = Y ± ( L ) ± Y − ± , which implies that ∀ k ∈ N ∗ , L k/α ± = Y ± ( L ) k/α ± Y − ± . We moreover have that ∀ β ∈ C ∗ , L β U = U L β which implies that L ± = Y ± ( L ) ± Y − ± = S ± S − ± Y ± ( L ) ± Y − ± S ± S − ± = S ± ( L ) ± S − ± and similarily L k/α ± = Y ± ( L ) k/α ± Y − ± = S ± ( L ) k/α ± S − ± We can now differentiate
UdU ± dt k = ( L ) k/α ± U ± = − S − ± dS ± dt k S − ± Y ± + S − ± dY ± dt k which implies that S ± ( L ) k/α ± S − ± = − dS ± dt k S − ± + dY ± dt k Y ± . By the way, (cid:16) L k/α ± (cid:17) D = dY ± dt k Y ± and hence dL ± dt k = dY ± ( L ) ± Y − ± dt k = dY ± dt k ( L ) ± Y − ± − Y ± ( L ) ± Y − ± dY ± dt k Y − ± = (cid:16) L k/α ± (cid:17) D Y ± ( L ) ± Y − ± − Y ± ( L ) ± Y − ± (cid:16) L k/α ± (cid:17) D = h(cid:16) L k/α ± (cid:17) D , Y ± ( L ) ± Y − ± i Gathering the ± parts, we get that L = L + + L − is a solution of (8) with initialcondition (6). Let us now deal with initial condition (7). In that case, L /α ∈ | D | + F Cl ( S , K ) = iǫ ( D ) (cid:18) ddx + F Cl ( S , K ) (cid:19) = iǫ ( D ) ddx + F Cl ( S , K ) . Let U be the dressing operator with respect to L along the lines of the previouscomputations, that we decompose into the U = U + + U − in the F Cl + ( S , K ) and F Cl − ( S , K ) − components. Then U + and U − decomposein the Mulase decomposition and we can re-construct two operators S and Y in F Cl ( S , K ) from this decomposition. We set L = Y L Y − = SL S − . Then, withthe same computation as before, dLdt k = dY L Y − dt k = dYdt k L Y − − Y L Y − ± dYdt k Y − = (cid:16) L k/α (cid:17) D Y L Y − − Y L Y − (cid:16) L k/α (cid:17) D = h(cid:16) L k/α (cid:17) D , Y L Y − i (cid:3) EALIZATION OF QUATERNIONIC KP HIERARCHY 27
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Inv. Math. , 143-178 (1984) and LAREMA, Université d’Angers,, 2 boulevard Lavoisier,, 49045 AngersCedex 1, France Lycée Jeanne d’Arc,, Avenue de Grande Bretagne,, 63000 Clermont-Ferrand,France ITEP, Theoretical Division, 25, Bol. Tcheremushkinskaya, 117259, Moscow,Russia IGAP (Institute for Geometry and Physics), Trieste, Italy
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