t^2
x'_s = (1 + tX + --- X.X + ...)x_s, (s = 1, 2, ..., n) (ii.)
1.2
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
t^2
F' = F + tX.F + --- X.X.F + ...
1.2
and
t'^2
F" = F' + t'X.F' + ---- X.X.F' + ...
1.2
is
(t + t')^2
F" = F + (t + t') X.F + ---------- 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, ..., e_r may be. Hence if e1t, e2t,
..., e_rt be replaced by a1, a2, ..., a_r 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 =
x^2[Pd]/[Pd]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.).
Relations between the infinitesimal operations of a finite continuous
group.
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 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, ..., X_r 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
__k=r
(X_iX_j) = \ c_(ijk)X_k,
/__k=1
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
c_(ijt) + c_(jit) = 0, \
|
and >
|
[Sigma][s](c_(jks)c_(ist) + c_(kis)c_(jst) + c_(ijs)c_(kst)) = 0 /
(iii.)
for all values of i, j, k and t.
Determination of the distinct types of continuous groups of a given
order.
The fundamental theorem of the theory of finite continuous groups is
now that these conditions, which are necessary in order that X1, X2,
..., X_r may generate, as infinitesimal operations, a continuous group
of order r, are also sufficient.
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, ..., X_r and Y1,
Y2, ..., Y_r can be chosen, so that in the relations
(X_iX_j) = [Sigma]c_(ijs)X_s, (Y_iY_j) = [Sigma]d_(ijs)Y_s,
the constants c_(ijs) and d_(ijs) are the same for all values of i, j
and s, the two groups are simply isomorphic, X_s and Y_s 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 r^3 quantities c_(ijs) which satisfy the relations
c_(ijt) + c_(ijt) = 0,
[Sigma] [c_(ijs)c_(skt) + c_(jks)c_(sit) + c_(kis)c_(sjt)] = 0.
s
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, ..., X_r may be replaced by
any r independent linear functions of themselves, and the c's will
then be transformed by a linear substitution containing r^2
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.