L.3.7 44

Even with a minus sign, however, Proposition 1 implies that the subspace »— (g) ‚

X(M) is a (¬nite dimensional) Lie subalgebra of the Lie algebra of all vector ¬elds on M.

Example: Linear Fractional Transformations. Consider the M¨bius action in-

o

troduced earlier of SL(2, R) on RP :

1

as + b

ab

·s= .

cd cs + d

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

01 10 00

x= , h= , y=

0 ’1

00 10

Thus, for example, the ¬‚ow Ψ» is given by

y

s

0 0 1 0

·s= ·s= = s ’ s2 t + · · · ,

Ψ» (t, s) = exp

y t 0 t 1 ts + 1

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

‚

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

‚s

The basic ODE existence theorem can be thought of as saying that every vector ¬eld

X ∈ X(M) arises as the “¬‚ow” of a “local” R-action on M. There is a generalization of

this to ¬nite dimensional subalgebras of X(M). To state it, we ¬rst de¬ne a local left action

of a Lie group G on a manifold M to be an open neighborhood U ‚ G — M of {e} — M

together with a smooth map »: U ’ M so that »(e, m) = m for all m ∈ M and so that

» a, »(b, m) = »(ab, m)

whenever this makes sense, i.e., whenever (b, m), (ab, m), and a, »(b, m) all lie in U.

It is easy to see that even a mere local Lie group action induces a map »— : g ’ 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 •: g ’ X(M) be a Lie algebra homomor-

phism. Then there exists a local left action (U, ») of G on M so that »— = ’•.

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 ‚ SL(2, R)—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 di¬erential 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 »— : g ’ X(M) where g is the Lie algebra of a Lie

group G, and a curve A: R ’ g, the ordinary di¬erential equation for a curve γ: R ’ M

γ (t) = »— A(t) γ(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) =

’a2 (t) ’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 ∈ GL(n, R) and B ∈ Rn

G= .

01

Then G acts on Rn by the standard a¬ne action:

A B

· x = Ax + B.

0 1

It is easy to verify that the inhomogeneous linear di¬erential 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 »: G—M ’ M relates

the right invariant vector ¬eld Yv to the vector ¬eld »— (v) on M. Despite its simplicity, it

has important consequences.

L.3.9 46

Proposition 3: If A: R ’ g is a curve in the Lie algebra of a Lie group G and S: R ’ 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 », the equation of Lie type

γ (t) = »— A(t) γ(t) ,

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

Lie™s Reduction Method. I now want to explain Lie™s method of analysing equa-

tions of Lie type. Suppose that »: G — M ’ M is a left action and that A: R ’ g is

a smooth curve. Suppose that we have found (by some method) a particular solution

γ: R ’ M of the equation of Lie type associated to A with γ(0) = m. Select a curve

g: R ’ G so that γ(t) = g(t) · 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 ’ Gm .

˜ ˜

I would like to choose h so that g is the fundamental solution of the Lie equation

˜

associated to A, i.e., so that

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

˜ ˜ ˜

Unwinding the de¬nitions, 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 ¬nd that h must satisfy the di¬erential equation

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

If we set

B(t) = Lg(t)’1 Rg(t) A(t) ’ g (t) ,

L.3.10 47

then B is clearly computable from g and A and hence may be regarded as known. Since

’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 ¬nd the fundamental solution of a Lie

equation for G when the particular solution with initial condition g(0) = m ∈ M is known,

it su¬ces to solve a Lie equation in Gm !

This observation is known as Lie™s method of reduction. It shows how knowledge of a

particular solution to a Lie equation simpli¬es the search for the general solution. (Note

that this is de¬nitely not true of general di¬erential equations.) Of course, Lie™s method

can be generalized. If one knows k particular solutions with initial values m1 , . . . , mk ∈ M,

then it is easy to see that one can reduce ¬nding the fundamental solution to ¬nding the

fundamental solution of a Lie equation in

Gm1 ,...,mk = Gm1 © Gm2 © · · · © 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) · 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

.

’1

vu

Thus, if we set, as usual,

a1 (t) a0 (t)

A(t) = ,

’a2 (t) ’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) = ’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

¬nd

b1 (t) 0 a1 (t) + a2 (t)s0 (t) 0

B(t) = = ,

b2 (t) ’b1 (t) ’a2 (t) ’a1 (t) ’ 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 („ )d„ ,

0

and, once u(t) has been found,

t

’1 2

v(t) = u(t) b2 („ ) u(„ ) d„.

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 ∈ GL(n, R) and B ∈ Rn

G= .

01

acting on Rn by the standard a¬ne action as before. If we embed Rn into Rn+1 by the

rule

x

x’ ,

1

then the standard a¬ne 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 ‚ G

of matrices of the form

In y

01

L.3.12 49

Thus, we choose

X(t) 0