be represented by Xi, (i = 1, 2, . . ., r), then the increment of F is
given by
(e1X1 + e2X2 + . . . + er Xr) Fδt.
When the equations (i.) defining the general operation of the group are given, the coefficients ∂ƒs/∂ai, which enter in these differential operators are functions of the variables which can be directly calculated.
The differential operator e1X1 + e2X2 + . . . + er Xr may then be regarded as defining the most general infinitesimal operation of the group. In fact, if it be for a moment represented by X, then (1 + δtX)F is the result of carrying out the infinitesimal operation on F; and by putting x1, x2, . . ., xn in turn for F, the actual infinitesimal operation is reproduced. By a very convenient, though perhaps hardly justifiable, phraseology this differential operator is itself spoken of as the general infinitesimal operation of the group. The sense in which this phraseology is to be understood will be made clear by the foregoing explanations.
We suppose now that the constants e1, e2, . . ., er have assigned values. Then the result of repeating the particular infinitesimal operation e1X1 + e2X2 + . . . + erXr or X an infinite number of times is some finite operation of the group. The effect of this finite operation on F may be directly calculated. In fact, if δt is the infinitesimal already introduced, then
| dF | = X·F, | d2F | = X·X·F, . . . |
| dt | dt2 |
Hence
| F′ = F + t | dF | + | t 2 | + | d2F | + . . . |
| dt | 1·2 | dt 2 |
| = F + tX·F + | t 2 | X·X·F + . . . |
| 1·2 |
It must, of course, be understood that in this analytical representation of the effect of the finite operation on F it is implied that t is taken sufficiently small to ensure the convergence of the (in general) infinite series.
When x1, x2, . . . are written in turn for F, the system of equations
| x′s = (1 + tX +t 21·2X·X + . . .)xs, (s＝1, 2, . . ., n) | (ii.) |
represent the finite operation completely. If t is here regarded as a parameter, this set of operations must in themselves constitute a group, since they arise by the repetition of a single infinitesimal operation. That this is really the case results immediately from noticing that the result of eliminating F′ between
| F′ = F + tX·F + | t 2 | X·X·F + . . . |
| 1·2 |
and
| F″ = F′ + t′X·F′ + | t′2 | X·X·F′ + . . . |
| 1·2 |
is
| F″ = F + (t + t ′) X·F + | (t + t ′)2 | X·X·F + . . . |
| 1·2 |
The group thus generated by the repetition of an infinitesimal operation is called a cyclical group; so that a continuous group contains a cyclical subgroup corresponding to each of its infinitesimal operations.
The system of equations (ii.) represents an operation of the group whatever the constants e1, e2, . . ., er may be. Hence if e1t, e2t, . . ., er t be replaced by a1, a2, . . ., ar the equations (ii.) represent a set of operations, depending on r parameters and belonging to the group. They must therefore be a form of the general equations for any operation of the group, and are equivalent to the equations (i.). The determination of the finite equations of a cyclical group, when the infinitesimal operation which generates it is given, will always depend on the integration of a set of simultaneous ordinary differential equations. As a very simple example we may consider the case in which the infinitesimal operation is given by X = x2∂/∂x, so that there is only a single variable. The relation between x′ and t is given by dx′/dt = x′2, with the condition that x′ = x when t = 0. This gives at once x′ = x/(1 − tx), which might also be obtained by the direct use of (ii.).
When the finite equations (i.) of a continuous group of order r are known, it has now been seen that the differential operator which defines the most general infinitesimal operation of the group can be directly constructed, and that it contains r arbitrary constants. This is equivalent to saying that Relations between the infinitesimal operations of a finite continuous group. the group contains r linearly independent infinitesimal operations; and that the most general infinitesimal operation is obtained by combining these linearly with constant coefficients. Moreover, when any r independent infinitesimal operations of the group are known, it has been seen how the general finite operation of the group may be calculated. This obviously suggests that it must be possible to define the group by means of its infinitesimal operations alone; and it is clear that such a definition would lend itself more readily to some applications (for instance, to the theory of differential equations) than the definition by means of the finite equations.
On the other hand, r arbitrarily given linear differential operators will not, in general, give rise to a finite continuous group of order r; and the question arises as to what conditions such a set of operators must satisfy in order that they may, in fact, be the independent infinitesimal operations of such a group.
If X, Y are two linear differential operators, XY − YX is also a linear differential operator. It is called the “combinant” of X and Y (Lie uses the expression Klammerausdruck) and is denoted by (XY). If X, Y, Z are any three linear differential operators the identity (known as Jacobi’s)
(X(YZ)) + (Y(ZX)) + (Z(XY)) = 0
holds between them. Now it may be shown that any continuous group of which X, Y are infinitesimal operations contains also (XY) among its infinitesimal operations. Hence if r linearly independent operations X1, X2, . . ., Xr give rise to a finite continuous group of order r, the combinant of each pair must be expressible linearly in terms of the r operations themselves: that is, there must be a system of relations
where the c’s are constants. Moreover, from Jacobi’s identity and the identity (XY) + (YX) = 0 it follows that the c’s are subject to the relations
| cijt + cjit = 0, | (iii.) | ||
| and | Σs (cjkscist + ckiscjst + cijsckst) = 0 |
for all values of i, j, k and t.
The fundamental theorem of the theory of finite continuous groups is now that these conditions, which are necessary in order that X1, X2, . . ., Xr may generate, as infinitesimal operations, a continuous group of order r, are also sufficient.Determination of the distinct types of continuous groups of a given order.
For the proof of this fundamental theorem see Lie’s works (cf. Lie-Engel, i. chap. 9; iii. chap. 25).
If two continuous groups of order r are such that, for each, a set of linearly independent infinitesimal operations X1, X2, . . ., Xr and Y1, Y2, . . ., Yr can be chosen, so that in the relations
(XiXj)＝Σcijs Xs, (YiYj) = Σ dijs Ys,
the constants cijs and dijs are the same for all values of i, j and s, the two groups are simply isomorphic, Xs and Ys being corresponding infinitesimal operations.
Two continuous groups of order r, whose infinitesimal operations obey the same system of equations (iii.), may be of very different form; for instance, the number of variables for the one may be different from that for the other. They are, however, said to be of the same type, in the sense that the laws according to which their operations combine are the same for both.
The problem of determining all distinct types of groups of order r is then contained in the purely algebraical problem of finding all the systems of r3 quantities cijs which satisfy the relations
cijt + cijt= 0,
Σs cijscskt + cjkscsit + ckiscsjt = 0.
for all values of i, j, k and t. To two distinct solutions of the algebraical problem, however, two distinct types of group will not necessarily correspond. In fact, X1, X2, . . ., Xr may be replaced by any r independent linear functions of themselves, and the c’s will then be transformed by a linear substitution containing r2 independent parameters. This, however, does not alter the type of group considered.
For a single parameter there is, of course, only one type of group, which has been called cyclical.
For a group of order two there is a single relation
(X1X2) = αX1 + βX2.
If α and β are not both zero, let α be finite. The relation may then be written (αX1 + βX2, α−1X2) = αX1 + βX2. Hence if αX1 + βX2 = X′1, and α−1X2 = X′2, then (X′1X′2) = X′1. There are, therefore, just two types of group of order two, the one given by the relation last written, and the other by (X1X2) = 0.
Lie has determined all distinct types of continuous groups of orders three or four; and all types of non-integrable groups (a term which will be explained immediately) of orders five and six (cf. Lie-Engel, iii. 713-744).
A problem of fundamental importance in connexion with any given continuous group is the determination of the self-conjugate subgroups which it contains. If X is an infinitesimal operation of a group, and Y any other, the general form of the infinitesimal operations which are Self-conjugate subgroups. Integrable groups. conjugate to X is
| X + t(XY) + | t 2 | ((XY)Y) + . . . . |
| 1.2 |
Any subgroup which contains all the operations conjugate to X must therefore contain all infinitesimal operations (XY), ((XY)Y), . . ., where for Y each infinitesimal operation of the group is taken in turn. Hence if X′1, X′2, . . ., X′s are s linearly independent operations of the group which generate a self-conjugate subgroup of order s, then for every infinitesimal operation Y of the group relations of the form
