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L.3.7 44
Even with a minus sign, however, Proposition 1 implies that the subspace О»в€— (g) вЉ‚
X(M) is a (п¬Ѓnite dimensional) Lie subalgebra of the Lie algebra of all vector п¬Ѓelds on M.

Example: Linear Fractional Transformations. Consider the MВЁbius action in-
o
troduced earlier of SL(2, R) on RP :
1

as + b
ab
В·s= .
cd cs + d

A basis for the Lie algebra sl(2, R) is

01 10 00
x= , h= , y=
0 в€’1
00 10

Thus, for example, the п¬‚ow ОЁО» is given by
y

s
0 0 1 0
В·s= В·s= = s в€’ s2 t + В· В· В· ,
ОЁО» (t, s) = exp
y t 0 t 1 ts + 1

so YyО» = в€’s2 в€‚/в€‚s. In fact, it is easy to see that, in general,

в€‚
О»в€— (a0 x + a1 h + a2 y) = (a0 + 2a1 s в€’ a2 s2 ) .
в€‚s

The basic ODE existence theorem can be thought of as saying that every vector п¬Ѓeld
X в€€ X(M) arises as the вЂњп¬‚owвЂќ of a вЂњlocalвЂќ R-action on M. There is a generalization of
this to п¬Ѓnite dimensional subalgebras of X(M). To state it, we п¬Ѓrst deп¬Ѓne a local left action
of a Lie group G on a manifold M to be an open neighborhood U вЉ‚ G Г— M of {e} Г— M
together with a smooth map О»: U в†’ M so that О»(e, m) = m for all m в€€ M and so that

О» a, О»(b, m) = О»(ab, m)

whenever this makes sense, i.e., whenever (b, m), (ab, m), and a, О»(b, m) all lie in U.
It is easy to see that even a mere local Lie group action induces a map О»в€— : g в†’ X(M)
as before. We can now state the following result, whose proof is left to the Exercises:

Proposition 2: Let G be a Lie group and let П•: g в†’ X(M) be a Lie algebra homomor-
phism. Then there exists a local left action (U, О») of G on M so that О»в€— = в€’П•.

For example, the linear fractional transformations of the last example could just as
easily been regarded as a local action of SL(2, R) on R, where the open set U вЉ‚ SL(2, R)Г—R
is just the set of pairs where cs + d = 0.

L.3.8 45
Equations of Lie type. Early in the theory of Lie groups, a special family of
ordinary diп¬Ђerential equations was singled out for study which generalized the theory of
linear equations and the Riccati equation. These have come to be known as equations of
Lie type. We are now going to describe this class.
Given a Lie algebra homomorphism О»в€— : g в†’ X(M) where g is the Lie algebra of a Lie
group G, and a curve A: R в†’ g, the ordinary diп¬Ђerential equation for a curve Оі: R в†’ M

Оі (t) = О»в€— A(t) Оі(t)

is known as an equation of Lie type.

Example: The Riccati equation. By our previous example, the classical Riccati
equation
2
s (t) = a0 (t) + 2a1 (t)s(t) + a2 (t) s(t)
is an equation of Lie type for the (local) linear fractional action of SL(2, R) on R. The
curve A is
a1 (t) a0 (t)
A(t) =
в€’a2 (t) в€’a1 (t)

Example: Linear Equations. Every linear equation is an equation of Lie type. Let G
be the matrix Lie subgroup of GL(n + 1, R),

AB
A в€€ GL(n, R) and B в€€ Rn
G= .
01

Then G acts on Rn by the standard aп¬ѓne action:

A B
В· x = Ax + B.
0 1

It is easy to verify that the inhomogeneous linear diп¬Ђerential equation

x (t) = a(t)x(t) + b(t)

is then a Lie equation, with
a(t) b(t)
A(t) = .
0 0

The following proposition follows from the fact that a left action О»: GГ—M в†’ M relates
the right invariant vector п¬Ѓeld Yv to the vector п¬Ѓeld О»в€— (v) on M. Despite its simplicity, it
has important consequences.

L.3.9 46
Proposition 3: If A: R в†’ g is a curve in the Lie algebra of a Lie group G and S: R в†’ G
is the solution to the equation S (t) = YA(t) (S(t)) with initial condition S(0) = e, then on
any manifold M endowed with a left G-action О», the equation of Lie type

Оі (t) = О»в€— A(t) Оі(t) ,

with initial condition Оі(0) = m has, as its solution, Оі(t) = S(t) В· m.

The solution S of Proposition 3 is often called the fundamental solution of the Lie
equation associated to A(t). The most classical example of this is the fundamental solution
of a linear system of equations:
x (t) = a(t)x(t)
where a is an n-by-n matrix of functions of t and x is to be a column of height n. In
ODE classes, we learn that every solution of this equation is of the form x(t) = X(t)x0
where X is the n-by-n matrix of functions of t which solves the equation X (t) = a(t)X(t)
with initial condition X(0) = In . Of course, this is a special case of Proposition 3 where
GL(n, R) acts on Rn via the standard left action described in Example 4.

LieвЂ™s Reduction Method. I now want to explain LieвЂ™s method of analysing equa-
tions of Lie type. Suppose that О»: G Г— M в†’ M is a left action and that A: R в†’ g is
a smooth curve. Suppose that we have found (by some method) a particular solution
Оі: R в†’ M of the equation of Lie type associated to A with Оі(0) = m. Select a curve
g: R в†’ G so that Оі(t) = g(t) В· m. Of course, this g will not, in general be unique, but any
other choice g will be of the form g(t) = g(t)h(t) where h: R в†’ Gm .
Лњ Лњ
I would like to choose h so that g is the fundamental solution of the Lie equation
Лњ
associated to A, i.e., so that

g (t) = YA(t) g (t) = Rg(t) A(t)
Лњ Лњ Лњ

Unwinding the deп¬Ѓnitions, it follows that h must satisfy

Rg(t)h(t) A(t) = Lg(t) h (t) + Rh(t) g (t)
so
Rh(t) Rg(t) A(t) = Lg(t) h (t) + Rh(t) g (t)

Solving for h (t), we п¬Ѓnd that h must satisfy the diп¬Ђerential equation

h (t) = Rh(t) Lg(t)в€’1 Rg(t) A(t) в€’ g (t) .

If we set
B(t) = Lg(t)в€’1 Rg(t) A(t) в€’ g (t) ,

L.3.10 47
then B is clearly computable from g and A and hence may be regarded as known. Since
в€’1
B = Rh(t) (h (t)) and since h is a curve in Gm , it follows that B must actually be a
curve in gm .
It follows that the equation

h (t) = Rh(t) B(t)

is a Lie equation for h. In other words in order to п¬Ѓnd the fundamental solution of a Lie
equation for G when the particular solution with initial condition g(0) = m в€€ M is known,
it suп¬ѓces to solve a Lie equation in Gm !
This observation is known as LieвЂ™s method of reduction. It shows how knowledge of a
particular solution to a Lie equation simpliп¬Ѓes the search for the general solution. (Note
that this is deп¬Ѓnitely not true of general diп¬Ђerential equations.) Of course, LieвЂ™s method
can be generalized. If one knows k particular solutions with initial values m1 , . . . , mk в€€ M,
then it is easy to see that one can reduce п¬Ѓnding the fundamental solution to п¬Ѓnding the
fundamental solution of a Lie equation in

Gm1 ,...,mk = Gm1 в€© Gm2 в€© В· В· В· в€© Gmk .

If one can arrange that this intersection is discrete, then one can explicitly compute a
fundamental solution which will then yield the general solution.

Example: The Riccati equation again. Consider the Riccati equation
2
s (t) = a0 (t) + 2a1 (t)s(t) + a2 (t) s(t)

and suppose that we know a particular solution s0 (t). Then let

1 s0 (t)
g(t) = ,
0 1

so that s0 (t) = g(t) В· 0 (we are using the linear fractional action of SL(2, R) on R). The
stabilizer of 0 is the subgroup G0 of matrices of the form:

u 0
.
в€’1
vu

Thus, if we set, as usual,
a1 (t) a0 (t)
A(t) = ,
в€’a2 (t) в€’a1 (t)
then the fundamental solution of S (t) = A(t)S(t) can be written in the form

u(t) 0
1 s0 (t)
S(t) = g(t)h(t) = в€’1
0 1 v(t) u(t)

L.3.11 48
Solving for the matrix B(t) (which we know will have values in the Lie algebra of G0 ), we
п¬Ѓnd
b1 (t) 0 a1 (t) + a2 (t)s0 (t) 0
B(t) = = ,
b2 (t) в€’b1 (t) в€’a2 (t) в€’a1 (t) в€’ a2 (t)s0 (t)
and the remaining equation to be solved is

h (t) = B(t)h(t),

which is solvable by quadratures in the usual way:
t
u(t) = exp b1 (П„ )dП„ ,
0

and, once u(t) has been found,
t
в€’1 2
v(t) = u(t) b2 (П„ ) u(П„ ) dП„.
0

Example: Linear Equations Again. Consider the general inhomogeneous n-by-n
system
x (t) = a(t)x(t) + b(t).
Let G be the matrix Lie subgroup of GL(n + 1, R),

AB
A в€€ GL(n, R) and B в€€ Rn
G= .
01

acting on Rn by the standard aп¬ѓne action as before. If we embed Rn into Rn+1 by the
rule
x
xв†’ ,
1
then the standard aп¬ѓne action of G on Rn extends to the standard linear action of G
on Rn+1 . Note that G leaves invariant the subspace xn+1 = 0, and solutions of the Lie
equation corresponding to
a(t) b(t)
A(t) =
0 0
which lie in this subspace are simply solutions to the homogeneous equation x (t) =
a(t)x(t). Suppose that we knew a basis for the homogeneous solutions, i.e., the fun-
damental solution to X (t) = a(x)X(t) with X(0) = In . This corresponds to knowing
the n particular solutions to the Lie equation on Rn+1 which have the initial conditions
e1 , . . . , en . The simultaneous stabilizer of all of these points in Rn+1 is the subgroup H вЉ‚ G
of matrices of the form
In y
01

L.3.12 49
Thus, we choose
X(t) 0
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