Part 3

To prevent any misapprehension as to the bearing of these very general results, it is well to point out explicitly that there are no limitations on the parameters of a continuous group as it has been defined above. They are to be regarded as taking in general complex values. If in the finite equations of a continuous group the imaginary symbol does not explicitly occur, the finite equations will usually define a group (in the general sense of the original definition) when both parameters and variables are limited to real values. Such a group is, in a certain sense, a continuous group; and such groups have been considered shortly by Lie (cf. Lie-Engel, iii. 360-392), who calls them _real_ continuous groups. To these real continuous groups the above statement as to the totality of simple groups does not apply; and indeed, in all probability, the number of types of _real_ simple continuous groups admits of no such complete enumeration. The effect of limitation to real transformations may be illustrated by considering the groups of projective transformations which change

x^2 + y^2 + z^2 - 1 = 0 and x^2 + y^2 - z^2 - 1 = 0

respectively into themselves. Since one of these quadrics is changed into the other by the imaginary transformation

x' = x, y' = y, z' = z[root](-1),

the general continuous groups which transform the two quadrics respectively into themselves are simply isomorphic. This is not, however, the case for the _real_ continuous groups. In fact, the second quadric has two real sets of generators; and therefore the real group which transforms it into itself has two self-conjugate subgroups, either of which leaves unchanged each of one set of generators. The first quadric having imaginary generators, no such self-conjugate subgroups can exist for the real group which transforms it into itself; and this real group is in fact simple.

The adjunct group.

Among the groups isomorphic with a given continuous group there is one of special importance which is known as the _adjunct_ group. This is a homogeneous linear group in a number of variables equal to the order of the group, whose infinitesimal operations are defined by the relations

[Pd] X_i=[Sigma] c_(ijs)x_i -------, (j = 1, 2, ..., r), i, s [Pd]x_s

where c_(ijs) are the often-used constants, which give the combinants of the infinitesimal operations in terms of the infinitesimal operations themselves.

That the r infinitesimal operations thus defined actually generate a group isomorphic with the given group is verified by forming their combinants. It is thus found that (X_pX_q) = [Sigma][s]c_(pqs)X_s. The X's, however, are not necessarily linearly independent. In fact, the sufficient condition that [Sigma][j]a_jX_j should be identically zero is that [Sigma][j]a_jc_(ijs) should vanish for all values of i and s. Hence if the equations [Sigma][j]a_jc_(ijs) = 0 for all values of i and s have r' linearly independent solutions, only r - r' of the X's are linearly independent, and the isomorphism of the two groups is multiple. If Y1, Y2, ..., Y_r are the infinitesimal operations of the given group, the equations

[Sigma] a_jc_(ijs) = 0, (s, i = 1, 2, ..., r) j

express the condition that the operations of the cyclical group generated by [Sigma][j]a_jY_i should be permutable with every operation of the group; in other words, that they should be self-conjugate operations. In the case supposed, therefore, the given group contains a subgroup of order r' each of whose operations is self-conjugate. The adjunct group of a given group will therefore be simply isomorphic with the group, unless the latter contains self-conjugate operations; and when this is the case the order of the adjunct will be less than that of the given group by the order of the subgroup formed of the self-conjugate operations.

Continuous groups of the line of the plane, and of three-dimensional space.

We have been thus far mainly concerned with the abstract theory of continuous groups, in which no distinction is made between two simply isomorphic groups. We proceed to discuss the classification and theory of groups when their form is regarded as essential; and this is a return to a more geometrical point of view.

It is natural to begin with the projective groups, which are the simplest in form and at the same time are of supreme importance in geometry. The general projective group of the straight line is the group of order three given by

ax + b x' = ------- cx + d'

where the parameters are the ratios of a, b, c, d. Since

x'3 - x'2 x' - x'1 x3 - x2 x - x1 --------- . -------- = ------- . ------ x'3 - x'1 x' - x'2 x3 - x1 x - x2

is an operation of the above form, the group is triply transitive. Every subgroup of order two leaves one point unchanged, and all such subgroups are conjugate. A cyclical subgroup leaves either two distinct points or two coincident points unchanged. A subgroup which either leaves two points unchanged or interchanges them is an example of a "mixed" group.

The analysis of the general projective group must obviously increase very rapidly in complexity, as the dimensions of the space to which it applies increase. This analysis has been completely carried out for the projective group of the plane, with the result of showing that there are thirty distinct types of subgroup. Excluding the general group itself, every one of these leaves either a point, a line, or a conic section unaltered. For space of three dimensions Lie has also carried out a similar investigation, but the results are extremely complicated. One general result of great importance at which Lie arrives in this connexion is that every projective group in space of three dimensions, other than the general group, leaves either a point, a curve, a surface or a linear complex unaltered.

Returning now to the case of a single variable, it can be shown that any finite continuous group in one variable is either cyclical or of order two or three, and that by a suitable transformation any such group may be changed into a projective group.

The genesis of an infinite as distinguished from a finite continuous group may be well illustrated by considering it in the case of a single variable. The infinitesimal operations of the projective group in one variable are d/dx, x(d/dx), x^2(d/dx). If these combined with x^3(d/dx) be taken as infinitesimal operations from which to generate a continuous group among the infinitesimal operations of the group, there must occur the combinant of x^2(d/dx) and x^3(d/dx). This is x^4(d/dx). The combinant of this and x^2(d/dx) is 2x^5(d/dx) and so on. Hence x^_r(d/dx), where r is any positive integer, is an infinitesimal operation of the group. The general infinitesimal operation of the group is therefore [f](x)(d/dx), where [f](x) is an arbitrary integral function of x.

In the classification of the groups, projective or non-projective of two or more variables, the distinction between primitive and imprimitive groups immediately presents itself. For groups of the plane the following question arises. Is there or is there not a singly-infinite family of curves [f](x, y) = C, where C is an arbitrary constant such that every operation of the group interchanges the curves of the family among themselves? In accordance with the previously given definition of imprimitivity, the group is called imprimitive or primitive according as such a set exists or not. In space of three dimensions there are two possibilities; namely, there may either be a singly infinite system of surfaces F(x, y, z) = C, which are interchanged among themselves by the operations of the group; or there may be a doubly-infinite system of curves G(x, y, z) = a, H(x, y, z) = b, which are so interchanged.

In regard to primitive groups Lie has shown that any primitive group of the plane can, by a suitably chosen transformation, be transformed into one of three definite types of projective groups; and that any primitive group of space of three dimensions can be transformed into one of eight definite types, which, however, cannot all be represented as projective groups in three dimensions.

The results which have been arrived at for imprimitive groups in two and three variables do not admit of any such simple statement.

Contact transformations.

We shall now explain the conception of contact-transformations and groups of contact-transformations. This conception, like that of continuous groups, owes its origin to Lie.

From a purely analytical point of view a contact-transformation may be defined as a point-transformation in 2n + 1 variables, z, x1, x2, ..., x_n, p1, p2, ..., p_n which leaves unaltered the equation dz - p1dx1 - p2dx2 - ... - p_ndx_n = 0. Such a definition as this, however, gives no direct clue to the geometrical properties of the transformation, nor does it explain the name given.

In dealing with contact-transformations we shall restrict ourselves to space of two or of three dimensions; and it will be necessary to begin with some purely geometrical considerations. An infinitesimal surface-element in space of three dimensions is completely specified, apart from its size, by its position and orientation. If x, y, z are the co-ordinates of some one point of the element, and if p, q, -1 give the ratios of the direction-cosines of its normal, x, y, z, p, q are five quantities which completely specify the element. There are, therefore, [oo]^5 surface elements in three-dimensional space. The surface-elements of a surface form a system of [oo]^2 elements, for there are [oo]^2 points on the surface, and at each a definite surface-element. The surface-elements of a curve form, again, a system of [oo]^2 elements, for there are [oo]^1 points on the curve, and at each [oo]^1 surface-elements containing the tangent to the curve at the point. Similarly the surface-elements which contain a given point clearly form a system of [oo]^2 elements. Now each of these systems of [oo]^2 surface-elements has the property that if (x, y, z, p, q) and (x + dx, y + dy, z + dz, p + dp, q + dq) are consecutive elements from any one of them, then dz - pdx - qdy = 0. In fact, for a system of the first kind dx, dy, dz are proportional to the direction-cosines of a tangent line at a point of the surface, and p, q, -1 are proportional to the direction-cosines of the normal. For a system of the second kind dx, dy, dz are proportional to the direction-cosines of a tangent to the curve, and p, q, -1 give the direction-cosines of the normal to a plane touching the curve; and for a system of the third kind dx, dy, dz are zero. Now the most general way in which a system of [oo]^2 surface-elements can be given is by three independent equations between x, y, z, p and q. If these equations do not contain p, q, they determine one or more (a finite number in any case) points in space, and the system of surface-elements consists of the elements containing these points; i.e. it consists of one or more systems of the third kind.

If the equations are such that two distinct equations independent of p and q can be derived from them, the points of the system of surface-elements lie on a curve. For such a system the equation dz - pdx - qdy = 0 will hold for each two consecutive elements only when the plane of each element touches the curve at its own point.

If the equations are such that only one equation independent of p and q can be derived from them, the points of the system of surface-elements lie on a surface. Again, for such a system the equation dz - pdx - qdy = 0 will hold for each two consecutive elements only when each element touches the surface at its own point. Hence, when all possible systems of [oo]^2 surface-elements in space are considered, the equation dz - pdx - qdy = 0 is characteristic of the three special types in which the elements belong, in the sense explained above, to a point or a curve or a surface.

Let us consider now the geometrical bearing of any transformation x' = [f]1(x, y, z, p, q), ..., q' = [f]5(x, y, z, p, q), of the five variables. It will interchange the surface-elements of space among themselves, and will change any system of [oo]^2 elements into another system of [oo]^2 elements. A special system, i.e. a system which belongs to a point, curve or surface, will not, however, in general be changed into another special system. The necessary and sufficient condition that a special system should always be changed into a special system is that the equation dz' - p'dx' - q'dy' = 0 should be a consequence of the equation dz - pdx - qdy = 0; or, in other words, that this latter equation should be invariant for the transformation.

When this condition is satisfied the transformation is such as to change the surface-elements of a surface in general into surface-elements of a surface, though in particular cases they may become the surface-elements of a curve or point; and similar statements may be made with respect to a curve or point. The transformation is therefore a veritable geometrical transformation in space of three dimensions. Moreover, two special systems of surface-elements which have an element in common are transformed into two new special systems with an element in common. Hence two curves or surfaces which touch each other are transformed into two new curves or surfaces which touch each other. It is this property which leads to the transformations in question being called contact-transformations. It will be noticed that an ordinary point-transformation is always a contact-transformation, but that a contact-transformation (in space of n dimensions) is not in general a point-transformation (in space of n dimensions), though it may always be regarded as a point-transformation in space of 2n + 1 dimensions. In the analogous theory for space of two dimensions a line-element, defined by (x, y, p), where 1 : p gives the direction-cosines of the line, takes the place of the surface-element; and a transformation of x, y and p which leaves the equation dy - pdx = 0 unchanged transforms the [oo]^1 line-elements, which belong to a curve, into [oo]^1 line-elements which again belong to a curve; while two curves which touch are transformed into two other curves which touch.

One of the simplest instances of a contact-transformation that can be given is the transformation by reciprocal polars. By this transformation a point P and a plane p passing through it are changed into a plane p' and a point P' upon it; i.e. the surface-element defined by P, p is changed into a definite surface-element defined by P', p'. The totality of surface-elements which belong to a (non-developable) surface is known from geometrical considerations to be changed into the totality which belongs to another (non-developable) surface. On the other hand, the totality of the surface-elements which belong to a curve is changed into another set which belong to a developable. The analytical formulae for this transformation, when the reciprocation is effected with respect to the paraboloid x^2 + y^2 - 2z = 0, are x' = p, y' = q, z' = px + qy - z, p' = x, q' = y. That this is, in fact, a contact-transformation is verified directly by noticing that dz' - p'dx' - q'dy' = -d(z - px - qy) - xdp - ydq = -(dz - pdx - qdy). A second simple example is that in which every surface-element is displaced, without change of orientation, normal to itself through a constant distance t. The analytical equations in this case are easily found in the form

pt qt x' = x + ---------------------, y' = y + --------------------, [root](1 + p^2 + q^2) [root](1 + p^2 + q^2)

t z' = z - ---------------------, [root](1 + p^2 + q^2)

p' = q, q' = q.

That this is a contact-transformation is seen geometrically by noticing that it changes a surface into a parallel surface. Every point is changed by it into a sphere of radius t, and when t is regarded as a parameter the equations define a cyclical group of contact-transformations.

The formal theory of continuous groups of contact-transformations is, of course, in no way distinct from the formal theory of continuous groups in general. On what may be called the geometrical side, the theory of groups of contact-transformations has been developed with very considerable detail in the second volume of Lie-Engel.

Applications of the theory of continuous groups.

To the manifold applications of the theory of continuous groups in various branches of pure and applied mathematics it is impossible here to refer in any detail. It must suffice to indicate a few of them very briefly. In some of the older theories a new point of view is obtained which presents the results in a fresh light, and suggests the natural generalization. As an example, the theory of the invariants of a binary form may be considered.

If in the form [f] = a0x^_n + na1x^(n-1)y + ... + a_ny^n, the variables be subjected to a homogeneous substitution

x' = [alpha]x + [beta]y, y' = [gamma]x + [delta]y, (i.)

and if the coefficients in the new form be represented by accenting the old coefficients, then

a'0 = a0[alpha]^n + a1n[alpha]^(n-1)[gamma] + ... + a_n[gamma]^n,\ | a'1 = a0[alpha]^(n-1)[beta] + a1_(n-1)[alpha]^(n-2)[beta][gamma] | + [alpha]^(n-1)[delta]} + ... + a_n[gamma]^(n-1)[delta], > (ii.) | a'_n = a0[beta]^n + a1n[beta]^(n-1)[delta] + ... + a_n[delta]^n; /

and this is a homogeneous linear substitution performed on the coefficients. The totality of the substitutions, (i.), for which [alpha][delta] - [beta][gamma] = 1, constitutes a continuous group of order 3, which is generated by the two infinitesimal transformations y([Pd]/[Pd]x) and x([Pd]/[Pd]y). Hence with the same limitations on [alpha], [beta], [gamma], [delta] the totality of the substitutions (ii.) forms a simply isomorphic continuous group of order 3, which is generated by the two infinitesimal transformations

[Pd] [Pd] [Pd] [Pd] a0 ------ + 2a1 ------ + 3a1 ------ + ... + na_(n-1) -------, [Pd]a1 [Pd]a2 [Pd]a3 [Pd]a_n

and

[Pd] [Pd] [Pd] [Pd] na1 ------ + (n - 1)a2 ------ + (n - 2)a3 ------ + ... + a_u ----------. [Pd]a0 [Pd]a1 [Pd]a2 [Pd]a_(u-1)

The invariants of the binary form, i.e. those functions of the coefficients which are unaltered by all homogeneous substitutions on x, y of determinant unity, are therefore identical with the functions of the coefficients which are invariant for the continuous group generated by the two infinitesimal operations last written. In other words, they are given by the common solutions of the differential equations

[Pd][f] [Pd][f] [Pd][f] a0 ------- + 2a1 ------- + 3a2 ------- + ... = 0, [Pd]a1 [Pd]a2 [Pd]a3

[Pd][f] [Pd][f] [Pd][f] na1 ------- + (n - 1)a2 ------- + (n - 2)a3 ------- + ... = 0. [Pd]a0 [Pd]a1 [Pd]a2

Both this result and the method by which it is arrived at are well known, but the point of view by which we pass from the transformation group of the variables to the isomorphic transformation group of the coefficients, and regard the invariants as invariants rather of the group than of the forms, is a new and a fruitful one.

The general theory of curvature of curves and surfaces may in a similar way be regarded as a theory of their invariants for the group of motions. That something more than a mere change of phraseology is here implied will be evident in dealing with minimum curves, i.e. with curves such that at every point of them dx^2 + dy^2 + dz^2 = 0. For such curves the ordinary theory of curvature has no meaning, but they nevertheless have invariant properties in regard to the group of motions.

The curvature and torsion of a curve, which are invariant for all transformations by the group of motions, are special instances of what are known as _differential invariants_. If [xi]([Pd]/[Pd]x) + [eta]([Pd]/[Pd]y) is the general infinitesimal transformation of a group of point-transformations in the plane, and if y1, y2, ... represent the successive differential coefficients of y, the infinitesimal transformation may be written in the extended form

[Pd] [Pd] [Pd] [Pd] [xi] ----- + [eta] ----- + [eta]1 ------ + [eta]2 ------ + ... [Pd]x [Pd]y [Pd]y1 [Pd]y2

where [eta]1[delta]t, [eta]2[delta]t, ... are the increments of y1, y2, .... By including a sufficient number of these variables the group must be intransitive in them, and must therefore have one or more invariants. Such invariants are known as differential invariants of the original group, being necessarily functions of the differential coefficients of the original variables. For groups of the plane it may be shown that not more than two of these differential invariants are independent, all others being formed from these by algebraical processes and differentiation. For groups of point-transformations in more than two variables there will be more than one set of differential invariants. For instance, with three variables, one may be regarded as independent and the other two as functions of it, or two as independent and the remaining one as a function. Corresponding to these two points of view, the differential invariants for a curve or for a surface will arise.

If a differential invariant of a continuous group of the plane be equated to zero, the resulting differential equation remains unaltered when the variables undergo any transformation of the group. Conversely, if an ordinary, differential equation [f](x, y, y1, y2, ...) = 0 admits the transformations of a continuous group, i.e. if the equation is unaltered when x and y undergo any transformation of the group, then [f](x, y, y1, y2, ...) or some multiple of it must be a differential invariant of the group. Hence it must be possible to find two independent differential invariants [alpha], [beta] of the group, such that when these are taken as variables the differential equation takes the form F([alpha], [beta], d[beta]/d[alpha], d^2[beta]/d[alpha]^2, ...) = 0. This equation in [alpha], [beta] will be of lower order than the original equation, and in general simpler to deal with. Supposing it solved in the form [beta] = [phi]([alpha]), where for [alpha], [beta] their values in terms of x, y, y1, y2, ... are written, this new equation, containing arbitrary constants, is necessarily again of lower order than the original equation. The integration of the original equation is thus divided into two steps. This will show how, in the case of an ordinary differential equation, the fact that the equation admits a continuous group of transformations may be taken advantage of for its integration.