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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 919538, 35 pages
The Lie Group in Infinite Dimension
Department of Mathematics, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic
Received 6 December 2010; Accepted 12 January 2011
Academic Editor: Miroslava Růžičková
Copyright © 2011 V. Tryhuk et al. 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.
A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local, smooth) action of a Lie group on infinite-dimensional space (a manifold modelled on ) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.
In the symmetry theory of differential equations, the generalized (or: higher-order, Lie-Bäcklund) infinitesimal symmetries where the coefficients are functions of independent variables , dependent variables and a finite number of jet variables belong to well-established concepts. However, in spite of this matter of fact, they cause an unpleasant feeling. Indeed, such vector fields as a rule do not generate any one-parameter group of transformations in the underlying infinite-order jet space since the relevant Lie system need not have any reasonable (locally unique) solution. Then is a mere formal concept [1–7] not related to any true transformations and the term “infinitesimal symmetry " is misleading, no -symmetries of differential equations in reality appear.
In order to clarify the situation, we consider one-parameter groups of local transformations in . We will see that they admit “finite-dimensional approximations" and as a byproduct, the relevant infinitesimal transformations may be exactly characterized by certain “finiteness requirements" of purely algebraical nature. With a little effort, the multidimensional groups can be easily involved, too. This result was briefly discussed in [8, page 243] and systematically mentioned at several places in monograph , but our aim is to make some details more explicit in order to prepare the necessary tools for systematic investigation of groups of generalized symmetries. We intend to continue our previous articles [10–13] where the algorithm for determination of all individual generalized symmetries was already proposed.
For the convenience of reader, let us transparently describe the crucial approximation result. We consider transformations (2.1) of a local one-parameter group in the space with coordinates . Equations (2.1) of transformations can be schematically represented by Figure 1(a).
We prove that in appropriate new coordinate system on , the same transformations become block triangular as in Figure 1(b). It follows that a certain hierarchy of finite-dimensional subspaces of is preserved which provides the “approximation" of . The infinitesimal transformation clearly preserves the same hierarchy which provides certain algebraical “finiteness" of .
If the primary space is moreover equipped with an appropriate structure, for example, the contact forms, it turns into the jet space and the results concerning the transformation groups on become the theory of higher-order symmetries of differential equations. Unlike the common point symmetries which occupy a number of voluminous monographs (see, e.g., [14, 15] and extensive references therein) this higher-order theory was not systematically investigated yet. We can mention only the isolated article  which involves a direct proof of the “finiteness requirements" for one-parameter groups (namely, the result of Lemma 5.4 below) with two particular examples and monograph  involving a theory of generalized infinitesimal symmetries in the formal sense.
Let us finally mention the intentions of this paper. In the classical theory of point or Lie's contact-symmetries of differential equations, the order of derivatives is preserved (Figure 2(a)). Then the common Lie's and Cartan's methods acting in finite dimensional spaces given ahead of calculations can be applied. On the other extremity, the generalized symmetries need not preserve the order (Figure 2(c)) and even any finite-dimensional space and then the common classical methods fail. For the favourable intermediate case of groups of generalized symmetries, the invariant finite-dimensional subspaces exist, however, they are not known in advance (Figure 2(b)). We believe that the classical methods can be appropriately adapted for the latter case, and this paper should be regarded as a modest preparation for this task.
2. Fundamental Approximation Results
Our reasonings will be carried out in the space with coordinates  and we introduce the structural family of all real-valued, locally defined and -smooth functions depending on a finite number of coordinates. In future, such functions will contain certain -smooth real parameters, too.
We are interested in (local) groups of transformations in defined by formulae where if the parameter is kept fixed. We suppose whenever it makes a sense. An open and common definition domain for all functions is tacitly supposed. (In more generality, a common definition domain for every finite number of functions is quite enough and the germ and sheaf terminology would be more adequate for our reasonings, alas, it looks rather clumsy.)
Definition 2.1. For every and , let be the subset of all composed functions where ; ; and is arbitrary -smooth function (of variables). In functions , variables are regarded as mere parameters.
Functions (2.3) will be considered on open subsets of where the rank of the Jacobi -matrix of functions locally attains the maximum (for appropriate choice of parameters). This rank and therefore the subset does not depend on as soon as is close enough to zero. This is supposed from now on and we may abbreviate .
We deal with highly nonlinear topics. Then the definition domains cannot be kept fixed in advance. Our results will be true locally, near generic points, on certain open everywhere dense subsets of the underlying space . With a little effort, the subsets can be exactly characterized, for example, by locally constant rank of matrices, functional independence, existence of implicit function, and so like. We follow the common practice and as a rule omit such routine details from now on.
Lemma 2.2 (approximation lemma). The following inclusion is true:
Proof. Clearly and therefore
Denoting by the rank of matrix (2.4), there exist basical functions such that . Then a function lies in if and only if is a composed function. In more detail is such a composed function if we choose given by (2.3). Parameters occurring in (2.3) are taken into account here. It follows that and analogously for the higher derivatives.
In particular, we also have for the choice in (2.9) whence The basical functions can be taken from the family of functions for appropriate choice of various values of . Functions (2.12) are enough as well even for a fixed value , for example, for , see Theorem 3.2 below.
Lemma 2.3. For any basical function, one has
Proof. implies and (2.9) may be applied with the choice and .
Summary 1. Coordinates ) ( were included into the subfamily which is transformed into itself by virtue of (2.13). So we have a one-parameter group acting on . One can even choose here and then, if is large enough, formulae (2.13) provide a “finite-dimensional approximation" of the primary mapping . The block-triangular structure of the infinite matrix of transformations mentioned in Section 1 appears if and the system of functions is succesively completed.
3. The Infinitesimal Approach
We introduce the vector field the infinitesimal transformation of group . Let us recall the celebrated Lie system In more explicit (and classical) transcription One can also check the general identity by a mere routine induction on .
Lemma 3.1 (finiteness lemma). For all , .
Theorem 3.2 (finiteness theorem). Every function admits (locally, near generic points) the representation in terms of a composed function where and is a -smooth function of a finite number of variables.
Proof. Let us temporarily denote
where the second equality follows from (3.4) with , . Then
by virtue of (3.4) with general .
If is large enough, there does exist an identity . Therefore by applying . This may be regarded as ordinary differential equation with initial values The solution expressed in terms of initial values reads in full detail. If is kept fixed, this is exactly the identity (3.6) for the particular case . The general case follows by a routine.
Definition 3.3. Let be the set of (local) vector fields such that every family of functions ( fixed but arbitrary) can be expressed in terms of a finite number of coordinates.
Remark 3.4. Neither nor as follows from simple examples. However, is a conical set (over ): if then for any . Easy direct proof may be omitted here.
Summary 2. If is of a group then all functions (; ) are included into family hence . The converse is clearly also true: every vector field generates a local Lie group since the Lie system (3.3) admits finite-dimensional approximations in spaces .
Let us finally reformulate the last sentence in terms of basical functions.
Theorem 3.5 (approximation theorem). Let be a vector field locally defined on and be a maximal functionally independent subset of the family of all functions Denoting , then the system may be regarded as a “finite-dimensional approximation" to the Lie system (3.3) of the one-parameter local group generated by .
In particular, assuming , then the the initial portion of the above system transparently demonstrates the approximation property.
4. On the Multiparameter Case
The following result does not bring much novelty and we omit the proof.
Theorem 4.1. Let be commuting local vector fields in the space . Then if and only if the vector fields () locally generate an abelian Lie group.
In full non-Abelian generality, let us consider a (local) multiparameter group formally given by the same equations (2.1) as above where are parameters close to the zero point . The rule (2.2) is generalized as where , and determine the composition of parameters. Appropriately adapting the space and the concept of basical functions , Lemma 2.2 holds true without any change.
Passing to the infinitesimal approach, we introduce vector fields which are of the group. We recall (without proof) the Lie equations  with the initial condition . Assuming linearly independent over , coefficients may be arbitrarily chosen and the solution always is a group transformation (the first fundamental theorem). If basical functions are inserted for , we have a finite-dimensional approximation which is self-contained in the sense that are composed functions in accordance with the definition of the basical functions.
Let us conversely consider a Lie algebra of local vector fields on the space . Let moreover uniformly in the sense that there is a universal space with for all . Then the Lie equations may be applied and we obtain reasonable finite-dimensional approximations.
Summary 3. Theorem 4.1 holds true even in the non-Abelian and multidimensional case if the inclusions are uniformly satisfied.
As yet we have closely simulated the primary one-parameter approach, however, the results are a little misleading: the uniformity requirement in Summary 3 may be completely omitted. This follows from the following result [9, page 30] needless here and therefore stated without proof.
Theorem 4.2. Let be a finite-dimensional submodule of the module of vector fields on such that . Then if and only if there exist generators (over ) of submodule that are lying in .
5. Symmetries of the Infinite-Order Jet Space
The previous results can be applied to the groups of generalized symmetries of partial differential equations. Alas, some additional technical tools cannot be easily explained at this place, see the concluding Section 11 below. So we restrict ourselves to the trivial differential equations, that is, to the groups of generalized symmetries in the total infinite-order jet space which do not require any additional preparations.
Let be the jet space of -dimensional submanifolds in [9–13]. We recall the familiar (local) jet coordinates Functions on are -smooth and depend on a finite number of coordinates. The jet coordinates serve as a mere technical tool. The true jet structure is given just by the module of contact forms or, equivalently, by the “orthogonal" module of formal derivatives Let us state useful formulae where denotes the Lie derivative.
Definition 5.1. We speak of a group of morphisms (5.5)of the jet structure if the inclusion holds true. We speak of a (universal) variation (5.6) of the jet structure if. If a variation (5.6) moreover generates a group, speaks of a (generalized or higher-order) infinitesimal symmetry of the jet structure.
So we intentionally distinguish between true infinitesimal transformations generating a group and the formal concepts; this point of view and the terminology are not commonly used in the current literature.
Remark 5.2. A few notes concerning this unorthodox terminology are useful here. In actual literature, the vector fields (5.6) are as a rule decomposed into the “trivial summand " and the so-called “evolutionary form " of the vector field , explicitly The summand is usually neglected in a certain sense [3–7] and the “essential" summand is identified with the evolutional system of partial differential equations (the finite subsystem with empty is enough here since the remaining part is a mere prolongation). This evolutional system is regarded as a “virtual flow" on the “space of solutions" , see [7, especially page 11]. In more generality, some differential constraints may be adjoint. However, in accordance with the ancient classical tradition, functions are just the variations. (There is only one novelty: in classical theory, are introduced only along a given solution while the vector fields are “universally" defined on the space.) In this “evolutionary approach", the properties of the primary vector field are utterly destroyed. It seems that the true sense of this approach lies in the applications to the topical soliton theory. However, then the evolutional system is always completed with boundary conditions and embedded into some normed functional spaces in order to ensure the existence of global “true flows". This is already quite a different story and we return to our topic.
In more explicit terms, morphisms (5.5) are characterized by the (implicit) recurrence where is supposed and vector field (5.6) is a variation if and only if Recurrence (5.9) easily follows from the inclusion and we omit the proof. Recurrence (5.10) follows from the identity and the inclusion . The obvious formula appearing on this occasion also is of a certain sense, see Theorem 5.5 and Section 10 below. It follows that the initial functions , , , (empty ) may be in principle arbitrarily prescribed in advance. This is the familiar prolongation procedure in the jet theory.
Let us recall that a vector field (5.6) generates a group (5.5) if and only if hence if and only if every family can be expressed in terms of a finite number of jet coordinates. We conclude with simple but practicable remark: due to jet structure, the infinite number of conditions (5.13) can be replaced by a finite number of requirements if is a variation.
Lemma 5.4. Let (5.6) be a variation of the jet structure. Then the inclusion is equivalent to any of the requirements () every family of functions can be expressed in terms of a finite number of jet coordinates,()every family of differential forms involves only a finite number of linearly independent terms,()every family of differential forms involves only a finite number of linearly independent terms.
Proof. Inclusion is defined by using the families (5.13) and this trivially implies where only the empty multi-indice is involved. Then implies by using the rule . Assuming , we may employ the commutative rule in order to verify identities of the kind and in full generality identities of the kind with unimportant coefficients, therefore follows. Finally obviously implies the primary requirement on the families (5.13).
This is not a whole story. The requirements can be expressed only in terms of the structural contact forms. With this final result, the algorithms [10–13] for determination of all individual morphisms can be closely simulated in order to obtain the algorithm for the determination of all groups of morphisms, see Section 10 below.
Theorem 5.5 (technical theorem). Let (5.6) be a variation of the jet space. Then if and only if every family involves only a finite number of linearly independent terms.
Some nontrivial preparation is needful for the proof. Let be a finite-dimensional module of 1-forms (on the space but the underlying space is irrelevant here). Let us consider vector fields such that for all functions . Let moreover be the module of all forms satisfying for all such . Then has a basis consisting of total differentials of certain functions (the Frobenius theorem), and there is a basis of module which can be expressed in terms of functions . Alternatively saying, (an appropriate basis of) the Pfaffian system () can be expressed only in terms of functions . This result frequently appears in Cartan's work, but we may refer only to [9, 18, 19] and to the appendix below for the proof.
Module is intrinsically related to : if a mapping preserves then preserves . In particular, assuming is true for a group . In terms of of the group , we have equivalent assertion and therefore for all . The preparation is done.
Proof. Let be the module generated by all differential forms (; ). Assuming finite dimension of module , we have module and clearly whence (). However involves both the differentials (see below) and the forms . Point of previous Lemma 5.4 implies . The converse is trivial.
In order to finish the proof, let us on the contrary assume that does not contain all differentials . Alternatively saying, the Pfaffian system () can be expressed in terms of certain functions such that does not imply . On the other hand, it follows clearly that maximal solutions of the Pfaffian system can be expressed only in terms of functions and therefore we do not need all independent variables . This is however a contradiction: the Pfaffian system consists of contact forms and involves the equations . All independent variables are needful if we deal with the common classical solutions .
The result can be rephrased as follows.
Theorem 5.6. Let be the submodule of all zeroth-order contact forms and be a variation of the jet structure. Then if and only if .
6. On the Multiparameter Case
Let us temporarily denote by the family of all infinitesimal variations (5.6) of the jet structure. Then , (), , and it follows that is an infinite-dimensional Lie algebra (coefficients in ). On the other hand, if and for certain then is a constant. (Briefly saying: the conical variations of the total jet space do not exist. We omit easy direct proof.) It follows that only the common Lie algebras over are engaged if we deal with morphisms of the jet spaces .
Theorem 6.1. Let be a finite-dimensional Lie subalgebra. Then if and only if there exists a basis of that is lying in .
7. The Order-Preserving Groups in Jet Space
Passing to particular examples from now on, we will briefly comment some well-known classical results for the sake of completeness.
Let be the submodule of all contact forms (sum with of the order at most. A morphism (5.5) and the infinitesimal variation (5.6) are called order preserving if respectively, for a certain (equivalently: for all , see Lemmas 9.1 and 9.2 below). Due to the fundamental Lie-Bäcklund theorem [1, 3, 6, 10–13], this is possible only in the pointwise case or in the Lie's contact transformation case. In quite explicit terms: assuming (7.1) then either functions , , , (empty in formulae (5.5) and (5.6) are functions only of the zeroth-order jet variables , or, in the second case, we have and all functions , , , , , contain only the zeroth- and first-order variables , , .
A somewhat paradoxically, short proofs of this fundamental result are not easily available in current literature. We recall a tricky approach here already applied in [10–13], to the case of the order-preserving morphisms. The approach is a little formally improved and appropriately adapted to the infinitesimal case.
Theorem 7.1 (infinitesimal Lie-Bäcklund). Let a variation preserve a submodule of contact forms of the order at most for a certain . Then and either is an infinitesimal point transformation or and is the infinitesimal Lie's contact transformation.
Proof. We suppose . Then therefore by virtue of Theorem 5.5. Moreover by virtue of Lemma 9.2 below. So we have Assuming , then (7.2) turns into the classical definition of Lie's infinitesimal contact transformation. Assume . In order to finish the proof we refer to the following result which implies that is indeed an infinitesimal point transformation.
Lemma 7.2. Let be a vector field on the jet space satisfying (7.2) and . Then are functions only of the point variables.
Proof. Let us introduce module of -forms generated by all forms of the kind
where . Clearly . The inclusions
are true by virtue of (7.2) and imply .
Module vanishes when restricted to certain hyperplanes, namely, just to the hyperplanes of the kind (use here). This is expressed by and it follows that Therefore again is such a hyperplane: (mod all and ). On the other hand, and it follows that , .
There is a vast literature devoted to the pointwise transformations and symmetries so that any additional comments are needless. On the other hand, the contact transformations are more involved and less popular. They explicitly appear on rather peculiar and dissimilar occasions in actual literature [20, 21]. However, in reality the groups of Lie contact transformations are latently involved in the classical calculus of variations and provide the core of the Hilbert-Weierstrass extremality theory of variational integrals.
8. Digression to the Calculus of Variations
We establish the following principle.
Theorem 8.1 (metatheorem). The geometries of nondegenerate local one-parameter groups of Lie contact transformations and of nondegenerate first-order one-dimensional variational integrals are identical. In particular, the orbits of a given group are extremals of appropriate and conversely.
Proof. The groups act in the jet space equipped with the contact module . Then the abbreviations
are possible. Let us recall the classical approach [22, 23]. The Lie contact transformations defined by certain formulae
preserve the Pfaffian equation or (equivalently) the submodule of zeroth-order contact forms. Explicit formulae are available in literature. We are interested in one-parameter local groups of transformations ) which are “nondegenerate" in a sense stated below and then the explicit formulae are not available yet. On the other hand, our with smooth Lagrangian Ł
to appear later, involves variables from quite other jet space with coordinates denoted (the independent variable), (the dependent variables) and higher-order jet variables like , and so on.
We are passing to the topic proper. Let us start in the space with groups. One can check that vector field (5.6) is infinitesimal if and only if where the function may be arbitrarily chosen.
“Hint: we have, by definition where , whence immediately , , if and formula (8.4) follows.”
Alas, the corresponding Lie system (not written here) is not much inspirational. Let us however consider a function implicitly defined by an equation . We may suppose that the transformed function satisfies the equation without any loss of generality. In infinitesimal terms However may be inserted here, and we have the crucial Jacobi equation (not involving ) which can be uniquely rewritten as the Hamilton-Jacobi equation in the “nondegenerate" case . Let us recall the characteristic curves [22, 23] of the equation given by the system The curves may be interpreted as the orbits of the group . (Hint: look at the well-known classical construction of the solution of the Cauchy problem [22, 23] in terms of the characteristics. The initial Cauchy data are transferred just along the characteristics, i.e., along the group orbits.) Assume moreover the additional condition . We may introduce variational integral (8.3) with the Lagrange function Ł given by the familiar identities with interrelations between variables , , of the space and variables , , of the space . Since (8.11) may be regarded as a Hamiltonian system for the extremals of , the metatheorem is clarified.
Remark 8.2. Let us recall the Mayer fields of extremals for the since they provide the true sense of the above construction. The familiar Poincaré-Cartan form
is restricted to appropriate subspace ) (the slope field) in order to become a total differential
of the action . We obtain the requirements , identical with (8.10). In geometrical terms: transformations of a hypersurface by means of group may be identified with the level sets ( of the action of a Mayer fields of extremals.
The last statement is in accordance with (8.11) where use the identifications (8.13) of coordinates. This is the classical definition of the action in a Mayer field. We have moreover clarified the additive nature of the level sets : roughly saying, the composition with provides (see Figure 3(c)) and this is caused by the additivity of the integral calculated along the orbits.
On this occasion, the wave enveloping approach to groups is also worth mentioning.
Lemma 8.3 (see [10–13]). Let be a function of variables. Assume that the system admits a unique solution by applying the implicit function theorem and analogously the system (where admits a certain solution Then , provides a Lie and , is the inverse.
In more generality, if function in Lemma 8.3 moreover depends on a parameter , we obtain a mapping which is a certain involving a parameter and the inverse . In favourable case (see below) this may be even a group. The geometrical sense is as follows. Equation with , kept fixed represents a wave in the space , (Figure 3(a)).
The total system provides the intersection (envelope) of infinitely close waves (Figure 3(b)) with the resulting transform, the focus point (or if the parameter is present). The reverse waves with the role of variables interchanged gives the inversion. Then the group property holds true if the waves can be composed (Figure 3(c)) within the parameters , , but this need not be in general the case.
Let us eventually deal with the condition ensuring the group composition property. Without loss of generality, we may consider the -depending wave If , are kept fixed, the previous results may be applied. We obtain a group if and only if the equation (8.10) holds true, therefore The existence of such function means that functions of dashed variables are functionally dependent whence The symmetry is not surprising here since the change provides the inverse mapping: equations are equivalent. In particular, it follows that and the wave corresponds to the Mayer central field of extremals.
Remark 8.4. A reasonable theory of Mayer fields of extremals and Hamilton-Jacobi equations can be developed also for the constrained variational integrals (the Lagrange problem) within the framework of jet spaces, that is, without the additional Lagrange multipliers [9, Chapter 3]. It follows that there do exist certain groups of generalized Lie's contact transformations with differential constraints.
9. On the Order-Destroying Groups in Jet Space
We recall that in the order-preserving case, the filtration of module is preserved (Figure 4(a)). It follows that certain invariant submodules are a priori prescribed which essentially restricts the store of the symmetries (the Lie-Bäcklund theorem). The order-destroying groups also preserve certain submodules of due to approximation results, however, they are not known in advance (Figure 4(b)) and appear after certain saturation (Figure 4(c)) described in technical theorem 5.1.
The saturation is in general a toilsome procedure. It may be simplified by applying two simple principles.
Lemma 9.1 (going-up lemma). Let a group of morphisms preserve a submodule . Then also the submodule is preserved.
Proof. We suppose . Then by using the commutative rule (5.17).
Lemma 9.2 (going-down lemma). Let the group of morphisms preserve a submodule . Let be the submodule of all satisfying (). Then is preserved, too.
Proof. Assume hence . Then hence and is preserved.
We are passing to illustrative examples.
Example 9.3. Let us consider the vector field (the variation of jet structure) see (5.6) and (5.10) for the particular case . Then () and the sufficient requirement () ensures , see of Lemma 5.4. We will deal with the linear case where is supposed. Then identically if and only if This may be expressed in terms of matrix equations or, in either of more geometrical transcriptions where is regarded as (a matrix of an) operator acting in -dimensional linear space and depending on parameters . We do not know explicit solutions in full generality, however, solutions such that does not depend on the parameters can be easily found (and need not be stated here). The same approach can be applied to the more general sufficient requirement (; fixed ) ensuring . If , the requirement is equivalent to the inclusion .
Example 9.4. Let us consider vector field (5.6) where . In more detail, we take Then and we have to deal with functions in order to ensure the inclusion . This is a difficult task. Let us therefore suppose Then () and where The second-order summand identically vanishes for the choice as follows by direct verification. Quite analogously It follows that all functions , can be expressed in terms of the finite family of functions (, (, ( and therefore .
Remark 9.5. On this occasion, let us briefly mention the groups generated by vector fields of the above examples. The Lie system of the vector field (9.4) and (9.5) reads
where we omit the prolongations. It is resolved by
as follows either by direct verification or, alternatively, from the property () which implies
Quite analogously, the Lie system of the vector field (9.10), (9.11), (9.15) reads
and may be completed with the equations
following from (9.16). This provides a classical self-contained system of ordinary differential equations where the common existence theorems can be applied.
The above Lie systems admit many nontrivial first integrals , that is, functions that are constant on the orbits of the group. Conditions may be interpreted as differential equations in the total jet space, and the above transformation groups turn into the external generalized symmetries of such differential equations, see Section 11 below.
10. Towards the Main Algorithm
Morphisms of the jet structure were defined by the property . The inverse exists if and only if for appropriate term of filtration (9.1). However and it follows that criterion (10.1) can be verified by repeated use of operators . In more detail, we start with equations with uncertain coefficients. Formulae (10.3) determine the module