- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

VolumeΒ 2011Β (2011), Article IDΒ 919538, 35 pages

http://dx.doi.org/10.1155/2011/919538

## The Lie Group in Infinite Dimension

Department of Mathematics, Faculty of Civil Engineering, Brno University of Technology, VeveΕΓ 331/95, 602 00 Brno, Czech Republic

Received 6 December 2010; Accepted 12 January 2011

Academic Editor: MiroslavaΒ RΕ―ΕΎiΔkovΓ‘

Copyright Β© 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.

#### Abstract

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.

#### 1. Preface

In the symmetry theory of differential equations, the *generalized* (*or*:* higher-order, Lie-BΓ€cklund*) *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 [1β7] not related to any true transformations and the term βinfinitesimal 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 βfinite-dimensional approximations" and as a byproduct, the relevant infinitesimal transformations may be exactly characterized by certain βfiniteness 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 [9], 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 [10β13] 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 βapproximation" of . The infinitesimal transformation clearly preserves the same hierarchy which provides certain algebraical βfiniteness" 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 [16] which involves a direct proof of the βfiniteness requirements" for one-parameter groups (namely, the result of Lemma 5.4 below) with two particular examples and monograph [7] 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 [9] 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 ; β; β 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 ββ 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 )ββ( 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 βfinite-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 , .*

*Proof. *Clearly
for any function (2.3) by virtue of (2.10): induction on .

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 ,ββ. 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 ββ ( 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 isββ of a group then all functions ββ(; ) 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 βfinite-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 ββ() 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 ,ββ 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 [17] 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 ββ on the space . Let moreover ββ*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 [9β13]. 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 *βorthogonal" module ** of formal derivatives *
Let us state useful formulae
where denotes the Lie derivative.

We are interested in (local) one-parameter groups of transformations given by certain formulae and in vector fields locally defined on the jet space ; see also (1.1) and (1.2).

*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 βtrivial summand " and the so-called βevolutionary form " of the vector field , explicitly
The summand is usually neglected in a certain sense [3β7] and the βessential" 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 βvirtual flow" on the βspace 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 βuniversally" defined on the space.) In this βevolutionary 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 βtrue 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.

*Remark 5.3. *Recurrence (5.10) for the variation can be succintly expressed by . This remarkable formula admits far going generalizations, see concluding Examples 11.3 and 11.4 below.

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 [10β13] 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 ββ*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 .*

The proof is elementary and may be omitted. Briefly saying, Theorem 4.2 (coefficients in ) turns into quite other and much easier Theorem 6.1 (coefficients 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-BΓ€cklund theorem [1, 3, 6, 10β13], 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 [10β13], 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-BΓ€cklund). *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 βnondegenerate" in a sense stated below and then the explicit formulae are not available yet. On the other hand, our with smooth Lagrangian Ε
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.

βHint: we have, by definition
where ,
whence immediately , , ββ if and formula (8.4) follows.β

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 βnondegenerate" 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 Ε 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 PoincarΓ©-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 [10β13]). *Let be a function of variables. Assume that the system admits a unique solution
**
by applying the implicit function theorem and analogously the system ββ (where admits a certain solution
**
Then , ββ 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 ββ*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.

*Summary 4. *Conditions (8.21) ensure the existence of equation (8.20) for the -wave (8.19) and therefore the group composition property of waves (8.19) in the nondegenerate case .

*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-BΓ€cklund 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 ββ() 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 ββ(; 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 ββ() 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 ββ() 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

We briefly recall the algorithm [10β13] for determination of all individual automorphisms of the jet space in order to compare it with the subsequent calculation of vector field .

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 . Then we search for lower-order contact forms, especially forms from , lying in with the use of (10.2). Such forms are ensured if certain *linear relations among coefficients exist*. The calculation is finished on a certain level and this is the *algebraic part* of the algorithm. With this favourable choice of coefficients , functions ,ββ (and therefore the invertible morphism ) can be determined by inspection of the bracket in (10.3). This is the *analytic part* of algorithm.

Let us turn to the infinitesimal theory. Then the main technical tool is the rule (5.17) in the following transcription:
or, when applied to basical forms
We are interested in vector fields . They satisfy the recurrence (5.10) together with requirements
for appropriate . Due to the recurrence (10.5) these requirements can be effectively investigated. In more detail, we start with equations
Formulae (10.7) determine module . Then, choosing , operator is to be repeatedly applied and requirements (10.6) provide certain *polynomial relations for the coefficients* by using (10.5). This is the *algebraical part* of the algorithm. With such coefficients available, functions ,ββ (and therefore the vector field ) can be determined by inspection of the bracket in (10.7) or, alternatively, with the use of formulae (5.12) for the particular case empty
This is the *analytic part* of the algorithm.

Altogether taken, the algorithm is not easy and the conviction [7, page 121] that the βexhaustive description of integrable -fields (fields in our notation) is given in [16]" is disputable. We can state only one optimistic result at this place.

Theorem 10.1. *The jet spaces do not admit any true generalized infinitesimal symmetries .*

*Proof. *We suppose and then (10.7) reads
where we state a summand of maximal order. Assuming , the Lie-BΓ€cklund theorem can be applied and we do not have the true generalized symmetry . Assuming , then
by using rule (10.5) where the last summand may be omitted. It follows that (10.6) is not satisfied hence .

*Example 10.2. * We discuss the simplest possible but still a nontrivial particular example. Assume , and . Let us abbreviate
Then, due to , requirement (10.6) reads
In particular (if ) we have (10.7) written here in the simplified notation
The next requirement () *implies the* (only seemingly) *stronger inclusion *
which already ensures (10.12) for all and therefore (easy). *We suppose* (10.14) *from now on*.

βHint for proof of (10.14): assuming (10.12) and moreover the equality
it follows that
and Lie-BΓ€cklund theorem can be applied whence ,ββ which we exclude. It follows that necessarily
On the other hand and the inclusion (10.14) follows.β

After this preparation, we are passing to the proper algebra. Clearly
by using the commutative rule (10.5). Due to βweaker" inclusion (10.12) with , we obtain identities
Omitting the trivial solution, they are satisfied if either
for appropriate factor (where and either or is supposed) or
We deal only with the (more interesting) identities (10.20) here. Then