0 1

as our initial guess and look for the fundamental solution in the form:

X(t) 0 In y(t)

S(t) = g(t)h(t) = .

0 1 0 1

Expanding the condition S (t) = A(t)S(t) and using the equation X (t) = a(t)X(t) then

reduces us to solving the equation

’1

y (t) = X(t) b(t),

which is easily solved by integration. The reader will probably recognize that this is

precisely the classical method of “variation of parameters”.

Solution by quadrature. This brings us to an interesting point: Just how hard is

it to compute the fundamental solution to a Lie equation of the form

γ (t) = Rγ(t) A(t) ?

One case where it is easy is if the Lie group is abelian. We have already seen that if T

is a connected abelian Lie group with Lie algebra t, then the exponential map exp: t ’ T

is a surjective homomorphism. It follows that the fundamental solution of the Lie equation

associated to A: R ’ t is given in the form

t

S(t) = exp A(„ ) d„

0

(Exercise: Why is this true?) Thus, the Lie equation for an abelian group is “solvable by

quadrature” in the classical sense.

Another instance where one can at least reduce the problem somewhat is when one has

a homomorphism φ: G ’ H and knows the fundamental solution SH to the Lie equation for

• —¦ A: R ’ h. In this case, SH is the particular solution (with initial condition SH (0) = e)

of the Lie equation on H associated to A by regarding φ as de¬ning a left action on H.

By Lie™s method of reduction, therefore, we are reduced to solving a Lie equation for the

group ker(φ) ‚ G.

Example. Suppose that G is connected and simply connected. Let g be its Lie algebra

and let [g, g] ‚ g be the linear subspace generated by all brackets of the form [x, y] where

x and y lie in g. Then, by the Exercises of Lecture 2, we know that [g, g] is an ideal in g

(called the commutator ideal of g). Moreover, the quotient algebra t = g/[g, g] is abelian.

Since G is connected and simply connected, Theorem 3 from Lecture 2 implies that

there is a Lie group homomorphism φ0 : G ’ T0 = t whose induced Lie algebra homo-

morphism •0 : g ’ t = g/[g, g] is just the canonical quotient mapping. From our previous

remarks, it follows that any Lie equation for G can be reduced, by one quadrature, to a

Lie equation for G1 = ker φ0 . It is not di¬cult to check that the group G1 constructed in

this argument is also connected and simply connected.

L.3.13 50

The desire to iterate this process leads to the following construction: De¬ne the

sequence {gk } of commutator ideals of g by the rules g0 = g and and gk+1 = [gk , gk ] for

k ≥ 0. Then we have the following result:

Proposition 4: Let G be a connected and simply connected Lie group for which the

sequence {gk } of commutator ideals satis¬es gN = (0) for some N > 0. Then any Lie

equation for G can be solved by a sequence of quadratures.

A Lie algebra with the property described in Proposition 4 is called “solvable”. For

example, the subalgebra of upper triangular matrices in gl(n, R) is solvable, as the reader

is invited to check.

While it may seem that solvability is a lot to ask of a Lie algebra, it turns out that

this property is surprisingly common. The reader can also check that, of all of the two

and three dimensional Lie algebras found in Lecture 2, only sl(2, R) and so(3) fail to be

solvable.

This (partly) explains why the Riccati equation holds such an important place in

the theory of ODE. In some sense, it is the ¬rst Lie equation which cannot be solved by

quadratures. (See the exercises for an interpretation and “proof” of this statement.)

In any case, the sequence of subalgebras {gk } eventually stabilizes at a subalgebra

gN whose Lie algebra satis¬es [gN , gN ] = gN . A Lie algebra g for which [g, g] = g is

called “perfect”. Our analysis of Lie equations shows that, by Lie™s reduction method,

we can, by quadrature alone, reduce the problem of solving Lie equations to the problem

of solving Lie equations associated to Lie groups with perfect algebras. Further analysis

of the relation between the structure of a Lie algebra and the solvability by quadratures

of any associated Lie equation leads to the development of the so-called Jordan-H¨lder

o

decomposition theorems, see [?].

L.3.14 51

Appendix: Lie™s Transformation Groups, I

When Lie began his study of symmetry groups in the nineteenth century, the modern

concepts of manifold theory were not available. Thus, the examples that he had to guide

him were de¬ned as “transformations in n variables” which were often, like the M¨bius o

transformations on the line or like conformal transformations in space, only de¬ned “almost

everywhere”. Thus, at ¬rst glance, it might appear that Lie™s concept of a “continuous

transformation group” should correspond to what we have de¬ned as a local Lie group

action.

However, it turns out that Lie had in mind a much more general concept. For Lie, a

set “ of local di¬eomorphisms in Rn formed a “continuous transformation group” if it was

closed under composition and inverse and moreover, the elements of “ were characterized

as the solutions of some system of di¬erential equations.

For example, the M¨bius group on the line could be characterized as the set “ of

o

(non-constant) solutions f(x) of the di¬erential equation

2

2f (x)f (x) ’ 3 f (x) = 0.

As another example, the “group” of area preserving transformations of the plane could be

characterized as the set of solutions f(x, y), g(x, y) to the equation

fx gy ’ gx fy ≡ 1,

while the “group” of holomorphic transformations of the plane

R2

(regarded as C) was the set of solutions f(x, y), g(x, y) to the equations

fx ’ gy = fy + gx = 0.

Notice a big di¬erence between the ¬rst example and the other two. In the ¬rst

example, there is only a 3-parameter family of local solutions and each of these solutions

patches together on RP1 = R ∪ {∞} to become an element of the global Lie group action

of SL(2, R) on RP1 . In the other two examples, there are many local solutions that cannot

be extended to the entire plane, much less any “completion”. Moreover in the volume

preserving example, it is clear that no ¬nite dimensional Lie group could ever contain all

of the globally de¬ned volume preserving transformations of the plane.

Lie regarded these latter two examples as “in¬nite continuous groups”. Nowadays, we

would call them “in¬nite dimensional pseudo-groups”. I will say more about this point of

view in an appendix to Lecture 6.

Since Lie did not have a group manifold to work with, he did not regard his “in¬nite

groups” as pathological. Instead of trying to ¬nd a global description of the groups, he

worked with what he called the “in¬nitesimal transformations” of “. We would say that,

L.3.15 52

for each of his groups “, he considered the space of vector ¬elds γ ‚ X(Rn ) whose (local)

¬‚ows were 1-parameter “subgroups” of “. For example, the in¬nitesimal transformations

associated to the area preserving transformations are the vector ¬elds

‚ ‚

X = f(x, y) + g(x, y)

‚x ‚y

which are divergence free, i.e., satisfy fx + gy = 0.

Lie “showed” that for any “continuous transformation group” “, the associated set

of vector ¬elds γ was actually closed under addition, scalar multiplication (by constants),

and, most signi¬cantly, the Lie bracket. (The reason for the quotes around “showed” is

that Lie was not careful to specify the nature of the di¬erential equations which he was

using to de¬ne his groups. Without adding some sort of constant rank or non-degeneracy

hypotheses, many of his proofs are incorrect.)

For Lie, every subalgebra L of the algebra X(Rn ) which could be characterized by

some system of pde was to be regarded the Lie algebra of some Lie group. Thus, rather

than classify actual groups (which might not really be groups because of domain problems),

Lie classi¬ed subalgebras of the algebra of vector ¬elds.

In the case that L was ¬nite dimensional, Lie actually proved that there was a “germ”

of a Lie group (in our sense) and a local Lie group action which generated this algebra of

vector ¬elds. This is Lie™s so-called Third Fundamental Theorem.

The case where L was in¬nite dimensional remained rather intractable. I will have

more to say about this in Lecture 6. For now, though, I want to stress that there is a sort

of analogue of actions for these “in¬nite dimensional Lie groups”.

For example, if M is a manifold and Diff(M) is the group of (global) di¬eomorphisms,

then we can regard the natural (evaluation) map »: Diff(M)—M ’ M given by »(φ, m) =

φ(m) as a faithful Lie group action. If M is compact, then every vector ¬eld is complete, so,

at least formally, the induced map »— : Tid Diff(M) ’ X(M) ought to be an isomorphism

of vector spaces. If our analogy with the ¬nite dimensional case is to hold up, »— must

reverse the Lie bracket.

Of course, since we have not de¬ned a smooth structure on Diff(M), it is not im-

mediately clear how to make sense of Tid Diff(M). I will prefer to proceed formally and

simply de¬ne the Lie algebra diff(M) of Diff(M) to be the vector space X(M) with the

Lie algebra bracket given by the negative of the vector ¬eld Lie bracket.

With this de¬nition, it follows that a left action »: G — M ’ M where G is ¬nite

dimensional can simply be regarded as a homomorphism Λ: G ’ Diff(M) inducing a

homomorphism of Lie algebras.

A modern treatment of this subject can be found in [SS].

L.3.16 53

Appendix: Connections and Curvature

In this appendix, I want brie¬‚y to describe the notions of connections and curvature

on principal bundles in the language that I will be using them in the examples in this

Lecture.

Let G be a Lie group with Lie algebra g and let ωG be the canonical g-valued, left-

invariant 1-form on G.

Principal Bundles. Let M be an n-manifold and let P be a principal right G-bundle

over M. Thus, P comes equipped with a submersion π: P ’ M and a free right action

ρ: P — G ’ P so that the ¬bers of π are the G-orbits of ρ.

The Gauge Group. The group Aut(P ) of automorphisms of P is, by de¬nition, the

set of di¬eomorphisms φ: P ’ P which are compatible with the two structure maps, i.e.,

π—¦φ=π ρg —¦ φ = φ —¦ ρg for all g ∈ G.

and

For reasons having to do with Physics, this group is nowadays referred to as the gauge

group of P . Of course, Aut(P ) is not a ¬nite dimensional Lie group, but it would have

been considered by Lie himself as a perfectly reasonable “continuous transformation group”

(although not a very interesting one for his purposes).

For any φ ∈ Aut(P ), there is a unique smooth map •: P ’ G which satis¬es φ(p) =

p · •(p). The identity ρg —¦ φ = φ —¦ ρg implies that • satis¬es •(p · g) = g ’1 •(p)g for all

g ∈ G. Conversely, any smooth map •: P ’ G satisfying this identity de¬nes an element

of Aut(P ). It follows that Aut(P ) is the space of sections of the bundle C(P ) = P —C G

where C: G — G ’ G is the conjugation action C(a, b) = aba’1 .

Moreover, it easily follows that the set of vector ¬elds on P whose ¬‚ows generate

1-parameter subgroups of Aut(P ) is identi¬able with the space of sections of the vector

bundle Ad(P ) = P —Ad g.

Connections. Let A(P ) denote the space of connections on P . Thus, an element

A ∈ A(P ) is, by de¬nition, a g-valued 1-form A on P with the following two properties:

(1) For any p ∈ P , we have ι— (A) = ωG where ιp : G ’ P is given by ιp (g) = p · g.

p

(2) For all g in G, we have ρg (A) = Ad(g ’1 )(A) where ρg : P ’ P is right action by g.

—

It follows from Property 1 that, for any connection A on P , we have A ρ— (x) = x

for all x ∈ g. It follows from Property 2 that Lρ— (x)A = ’[x, A] for all x ∈ g.

If A0 and A1 are connections on P , then it follows from Property 1 that the di¬erence

± = A1 ’ A0 is a g-valued 1-form which is “semi-basic” in the sense that ±(v) = 0 for all

v ∈ ker π . Moreover, Property 2 implies that ± satis¬es ρ— (±) = Ad(g ’1 )(±). Conversely,

g

if ± is any g-valued 1-form on P satisfying these latter two properties and A ∈ A(P ) is a

connection, then A + ± is also a connection. It is easy to see that a 1-form ± with these

two properties can be regarded as a 1-form on M with values in Ad(P ).

Thus, A(P ) is an a¬ne space modeled on the vector space A1 Ad(P ) . In particular,

if we regard A(P ) as an “in¬nite dimensional manifold”, the tangent space TA A(P ) at any

point A is naturally isomorphic to A1 Ad(P ) .

L.3.17 54

Curvature. The curvature of a connection A is the 2-form FA = dA + 1 [A, A]. From

2

our formulas above, it follows that

ρ— (x) FA = ρ— (x) dA + [x, A] = Lρ— (x)A + [x, A] = 0.

Since the vector ¬elds ρ— (x) span the vertical tangent spaces of P , it follows that FA is a

“semi-basic” 2-form (with values in g). Moreover, the Ad-equivariance of A implies that

ρ— (FA ) = Ad(g ’1 )(FA ). Thus, FA may be regarded as a section of the bundle of 2-forms

g

on M with values in the bundle Ad(P ).

The group Aut(P ) acts naturally on the right on A(P ) via pullback: A · φ = φ— (A).

In terms of the corresponding map •: P ’ G, we have

A · φ = •— (ωG ) + Ad •’1 (A).

It follows by direct computation that FA·φ = φ— (FA ) = Ad •’1 (FA ).

We say that A is ¬‚at if FA = 0. It is an elementary ode result that A is ¬‚at if and

only if, for every m ∈ M, there exists an open neighborhood U of m and a smooth map

„ : π ’1 (U) ’ G which satis¬es „ (p · g) = „ (p)g and „ — (ωG ) = A|U . In other words A is

¬‚at if and only if the bundle-with-connection (P, A) is locally di¬eomorphic to the trivial

bundle-with-connection (M — G, ωG ).

Covariant Di¬erentiation. The space Ap Ad(P ) of p-forms on M with values

in Ad(P ) can be identi¬ed with the space of g-valued, p-forms β on P which are both

semi-basic and Ad-equivariant (i.e., ρ— (β) = Ad(g ’1 )(β) for all g ∈ G). Given such a

g

form β, the expression dβ + [A, β] is easily seen to be a g-valued (p+1)-form on P which

is also semi-basic and Ad-equivariant. It follows that this de¬nes a ¬rst-order di¬erential

operator

dA : Ap Ad(P ) ’ Ap+1 Ad(P )

called covariant di¬erentiation with respect to A. It is elementary to check that

dA dA β = [FA , β] = ad(FA )(β).

Thus, for a ¬‚at connection, A— (Ad(P )), dA forms a complex over M.

We also have the Bianchi identity dA FA = 0.

For some, “covariant di¬erentiation” means only dA : A0 (Ad(P )) ’ A1 (Ad(P )).

Horizontal Lifts and Holonomy. Let A be a connection on P . If γ: [0, 1] ’ M is

a C 1 curve and p ∈ π ’1 γ(0) is chosen, then there exists a unique C 1 curve γ : [0, 1] ’ P

˜

which both “lifts” γ in the sense that γ = π —¦ γ and also satis¬es the di¬erential equation

˜

γ — (A) = 0.

˜

(To see this, ¬rst choose any lift γ : [0, 1] ’ P which satis¬es γ (0) = p. Then the

¯ ¯

desired lifting will then be given by γ (t) = γ (t) · g(t) where g: [0, 1] ’ G is the solution of

˜ ¯

the Lie equation g (t) = ’Rg(t) A(¯ (t)) satisfying the initial condition g(0) = e.)

γ

L.3.18 55