ON ANTIPLECTIC PAIRS IN THE HAMILTONIAN FORMALISM OF EVOLUTION EQUATIONS .sx By GEORGE WILSON .sx Introduction .sx IT is well known that evolutionary systems of the type of the Korteweg-de Vries ( KdV ) equation can ( under suitable boundary conditions ) be regarded as examples of infinite dimensional integrable Hamiltonian systems .sx Recently ( see [9] ) I have argued that these systems typically occur as part of a pair of equations related by a transformation of the dependent variables .sx The relationship between the two equations of the pair is analogous to the relationship between a Lie group and its Lie algebra .sx From the Hamiltonian viewpoint , these pairs have a curious feature :sx one of the flows in question takes place on a ( slightly degenerate ) symplectic manifold , the other on a Poisson manifold .sx The purpose of this paper is to lay the foundations for that part of the theory of such pairs that is independent of the boundary conditions , and hence of a purely algebraic nature .sx The appropriate language for this is what Gel'fand and Dikii call the formal calculus of variations ( see [3]) .sx We begin by reviewing the relevant background material .sx In the literature on finite dimensional Hamiltonian systems , one can observe some mild disagreement as to whether the notion of a symplectic manifold or a Poisson manifold is the better starting point .sx We recall first the latter notion .sx Let M be a smooth manifold , l a skew tensor of type ( 2,0 ) on M. We can think of l as a skew form on the cotangent bundle T*M ; or , equivalently , we can consider the corresponding homomorphism formula .sx Thus l # assigns to each 1-form on M a vector field .sx Let A denote the algebra of all smooth real valued functions on M. Then for each f unch A we have the 1-form df , and hence the vector field l # ( df ) :sx we shall think of this vector field as a derivation of A and denote it by delta f .sx Thus we have , for any f , g unch A .sx formula .sx The function formula is also denoted by f , g and called the Poisson bracket of f and g. It is evidently skew , since l is .sx If in addition we have , for every f , g unch A , .sx formula&caption .sx or , equivalently , if the bracket satisfies the Jacobi identity and the map formula is a homomorphism of Lie algebras , then we say that ( M , l ) is a Poisson manifold .sx The condition ( 1.1 ) translates into a quadratic condition on the tensor l , namely , the vanishing of a certain skew 3-tensor , the Schouten-Nijenhuis bracket of l with itself :sx see , for example , [ 4 , 8 ] .sx If now in addition to all the above l is non-degenerate , that is , l # is an isomorphism , we can consider the inverse isomorphism formula , and the corresponding non-degenerate 2-form lambda .sx The pair ( M , lambda ) is then called a symplectic manifold .sx One advantage of formulating the theory in terms of lambda , rather than the inverse tensor l , is that the crucial condition ( 1.1 ) becomes a very simple linear condition on the 2-form lambda , namely that it should be closed :sx d lambda = 0 .sx Nevertheless , from this point of view , a symplectic manifold is just a special case of a Poisson manifold .sx On the other hand , any Poisson manifold ( M , l ) comes equipped with a kind of foliation by symplectic manifolds in which the tangent spaces to the leaves ( which in general may not all have the same dimension ) make up the image of the map l # .sx All the Hamiltonian flows on M ( that is , the integral curves of the vector fields delta f ) preserve the symplectic leaves of this foliation .sx The effect of all this is that it usually makes little difference whether we regard lambda or l as the more fundamental object .sx Before passing to the infinite dimensional case , let us observe that all the discussion above could be formulated in more algebraic language in terms of the algebra A of functions on M ; this would be particularly appropriate in the variant of the theory where M was an affine algebraic variety and A the algebra of regular functions on M. Instead of the cotangent bundle to M we should then speak of the universal A -module with derivation OMEGA ; the tensor l above would become a skew bilinear form formula , and a 2-form would be a skew form on the dual A -module OMEGA * .sx Now let us consider the prototype example of the infinite dimensional systems that we want to study , the KdV equation .sx formula&caption .sx We want to regard this equation as defining a flow on some infinite dimensional manifold M of functions u( x ) ( we prescribe the initial value u( x , 0 ) and ( 1.2 ) then determines u( x , t )) .sx For that we have to fix suitable boundary conditions ; for example , we can take M to be the space of all smooth periodic functions u(x ) ( of some fixed period) .sx The interpretation of the KdV equation as a Hamiltonian flow on M is based on the fact that ( 1.2 ) can be written in the Hamiltonian form .sx formula&caption .sx where .sx formula&caption .sx and formula ; here we write formula , and formula is the Euler-Lagrange operator :sx in general , if f is a function of u , u x , u xx , .sx .. then .sx formula&caption .sx The justification for calling ( 1.3 ) 'Hamiltonian' is as follows .sx On the vector space M we have the usual L 2 inner product formula ( where the integral is taken over a period ) which identifies M with a subspace of its dual M* ; we call this subspace the 'smooth part' of M* .sx If formula is the function .sx formula .sx one easily finds that the exterior derivative dH belongs to the smooth part of M* , and that its value at a point u unch M , regarded as an element of formula , is just formula .sx Thus if we regard the skew differential operator l in ( 1.4 ) as analogous to the map l # in our discussion of finite dimensional Poisson manifolds , then ( provided we can check the vital condition ( 1.1) ) it is clear that ( 1.3 ) is indeed the vector field on the Poisson manifold M corresponding to the function H. We shall not pursue here the question of making this discussion rigorous , but shall pass at once to the algebraic formulation of the theory .sx By that I do not mean simply replacing the infinite dimensional manifold M by its algebra of functions :sx that would achieve no simplification , and would ignore two important features of our situation .sx The first is that our manifold M is not just an abstract manifold , but consists of functions of x. Thus on M we have a distinguished vector field formula , and we are interested only in vector fields formula that commute with delta .sx Still more significant is the remark that , because ( 1.3 ) involves only the 'density' f , and not the function formula , the boundary conditions used in our discussion are for many purposes irrelevant :sx for example , if we were to replace M by a suitable space of functions on the whole line , decaying at infinity , a large part of the theory would remain unaffected .sx The basic question of whether ( 1.1 ) is satisfied belongs to this purely local part of the theory .sx For these reasons we follow the influential paper [3] of Gel'fand and Dikii , and proceed as follows .sx Let formula be the algebra of differential polynomials in u :sx that is , polynomials in the infinite set of variables formulae ( we write formulae , ) .sx We equip A with the unique derivation delta such that formula .sx We now try to think of the differential algebra ( A , delta ) as the algebra of functions on the manifold M ( although it is not ) , and imitate the algebraic version of the finite dimensional theory .sx The role of the vector fields on M is played by the Lie algebra of evolutionary derivations of A , that is , derivations commuting with delta .sx Clearly such a derivation of A is entirely determined by its value on u , and this value may be chosen arbitrarily .sx Thus equation ( 1.2 ) ( or ( 1.3) ) can be interpreted as defining an evolutionary derivation of A. Of course , this way of studying the algebraic properties of an evolutionary system is quite classical , and has long been a standard point of view in discussing the local conservation laws of such systems ; however , the fact that much of the Hamiltonian formalism too can be described within this framework seems first to have been made clear in [3] .sx Given the skew differential operator l , the basic construction of the theory now assigns to each f unch A the unique evolutionary derivation delta f of A such that formula is the right-hand side of ( ) .sx We then call l Hamiltonian if a suitably formulated version of ( 1.1 ) holds .sx In this algebraic context the idea of considering the inverse of the differential operator l seems very unnatural ; and it is doubtful if we could make any sense of discussing the symplectic leaves of a foliation defined by l. Therefore , the following attitude towards the Hamiltonian formalism has become standard :sx we should simply accept that l defines a Poisson structure in the way sketched above , and abandon any idea of relating it to a symplectic structure , as we do in the classical case .sx In [9] , however , I have argued that this may not be the best attitude .sx For the KdV equation and its generalizations , the operator l , although not invertible , is in a certain sense obtained by inverting a 2-form lambda ; moreover , it is lambda that is the more fundamental object .sx The justification for this last claim is that lambda is invariant with respect to a symmetry group G ( for the KdV equation G is PSL ( 2) ) and l is obtained from lambda essentially by passing to the quotient by G. The construction of lambda from l is less straightforward , especially since in practice the group G is not given to us in advance .sx As an example , the 2-form lambda corresponding to the l in ( 1.4 ) is .sx formula&caption .sx where the variables psi and u are related by .sx formula&caption .sx ( this last expression is the Schwarzian derivative of psi :sx it is invariant with respect to the group G = PSL ( 2 ) acting on psi by linear fractional transformations) .sx We shall call such a pair of variables ( psi , u ) with their associated skew operators ( lambda , l ) an antiplectic pair .sx As we shall see , a consequence of the existence of lambda is that each equation of the form ( 1.3 ) for u is implied by a symplectic G-invariant equation for psi .sx For example , the KdV equation ( 1.2 ) follows from the 'Ur-KdV' equation .sx formula&caption .sx for psi .sx This equation is symplectic in the sense that it can formally be written as .sx formula&caption .sx where H = u 2 is the Hamiltonian for the KdV equation .sx It is a basic part of the machinery that we are going to discuss that the appearance of lambda -1 here does not actually require us to integrate anything :sx in fact for any Hamiltonian H depending only on u , u x , .sx .. we have .sx formula .sx so that the lambda here cancels the lambda -1 in ( 1.9 ) to give ( ) .sx This example is considered in more detail in [9] , and in section 7 below .sx The main purpose of the present paper , then , is to give a precise formulation of the notion of an antiplectic pair , and in particular to clarify the process by which the type of the G -invariant tensor lambda reverses when we divide out by G. The paper is arranged as follows .sx Section 2 is preliminary , and is intended for reference only .sx It sets out some elementary facts ( and conventions ) about modules over rings with an anti-involution .sx The motivation for that is the following :sx the basic derivation delta of the algebra A acts also on the universal A -module OMEGA that plays the role , in the algebraic theory , of the contangent bundle T*M .sx Thus OMEGA acquires a structure of left module over the algebra R of ( ordinary ) differential operators with coefficients in A :sx this structure plays an important role in our version of the formalism .sx On R we have the anti-involution formula , where L* is the formal adjoint of L. This makes possible certain constructions that could not be performed for a general non-commutative ring R.