8. Show that, for any Lie group G, the mappings La satisfy

’1

La (b) = Lab(e) —¦ (Lb (e)) .

where La (b): Tb G ’ Tab G. (This shows that the e¬ect of left translation is completely

determined by what it does at e.) State and prove a similar formula for the mappings Ra .

9. Let (G, µ) be a Lie group. Using the canonical identi¬cation T(a,b)(G—G) = Ta G•Tb G,

prove the formula

µ (a, b)(v, w) = Rb (a)(v) + La (b)(w)

for all v ∈ Ta G and w ∈ Tb G.

10. Complete the proof of Proposition 3 by explicitly exhibiting the map c as a composition

of known smooth maps. (Hint: if f: X ’ Y is smooth, then f : T X ’ T Y is also smooth.)

11. Show that, for any v ∈ g, the left-invariant vector ¬eld Xv is indeed smooth. Also prove

the ¬rst statement in Proposition 4. (Hint: Use Ψ to write the mapping Xv : G ’ T G as

a composition of smooth maps. Show that the assignment v ’ Xv is linear. Finally, show

that if a left-invariant vector ¬eld on G vanishes anywhere, then it vanishes identically.)

12. Show that exp: g ’ G is indeed smooth and that exp : g ’ g is the identity mapping.

(Hint: Write down a smooth vector ¬eld Y on g — G such that the integral curves of Y are

of the form γ(t) = (v0 , a0 etv0 ). Now use the ¬‚ow of Y ,

Ψ: R — g — G ’ g — G,

to write exp as the composition of smooth maps.)

13. Show that, for the homomorphism det: GL(n, R) ’ R• , we have det (In )(x) = tr(x),

where tr denotes the trace function. Conclude, using Theorem 1 that, for any matrix a,

det(ea ) = etr(a) .

14. Prove that, for any g ∈ G and any x ∈ g, we have the identity

g exp(x) g ’1 = exp Ad(g)(x) .

(Hint: Replace x by tx in the above formula and consider Proposition 5.) Use this to show

that tr exp(x) ≥ ’2 for all x ∈ sl(2, R). Conclude that exp: sl(2, R) ’ SL(2, R) is not

surjective. (Hint: show that every x ∈ sl(2, R) is of the form gyg ’1 for some g ∈ SL(2, R)

and some y which is one of the matrices

0 ±1 ’»

» 0 0

, , or , (» > 0).

’»

00 0 » 0

Also, remember that tr aba’1 = tr(b).)

E.2.2 34

15. Using Theorem 1, show that if H1 and H2 are Lie subgroups of G, then H1 © H2 is

also a Lie subgroup of G. (Hint: What should the Lie algebra of this intersection be? Be

careful: H1 © H2 might have countably many distinct components even if H1 and H2 are

connected!)

16. For any skew-commutative algebra (g, [, ]), we de¬ne the map ad: g ’ End(g) by

ad(x)(y) = [x, y]. Verify that the validity of the Jacobi identity [ad(x), ad(y)] = ad([x, y])

(where, as usual, the bracket on End(g) is the commutator) is equivalent to the validity of

the identity

[[x, y], z] + [[y, z], x] + [[z, x], y] = 0

for all x, y, z ∈ g.

17. Show that, as » ∈ R varies, all of the groups

ab

a ∈ R+ , b ∈ R

G» =

0 a»

with » = 1 are isomorphic, but are not conjugate in GL(2, R). What happens when » = 1?

18. Show that a connected Lie group G is abelian if and only if its Lie algebra satis¬es

[x, y] = 0 for all x, y ∈ g. Conclude that a connected abelian Lie group of dimension n is

isomorphic to Rn /Zd where Zd is some discrete subgroup of rank d ¤ n. (Hint: To show

“G abelian” implies “g abelian”, look at how [, ] was de¬ned. To prove the converse, use

Theorem 3 to construct a surjective homomorphism φ: Rn ’ G with discrete kernel.)

˜

19. (Covering Spaces of Lie groups.) Let G be a connected Lie group and let π: G ’ G be

˜

the universal covering space of G. (Recall that the points of G can be regarded as the space

of ¬xed-endpoint homotopy classes of continuous maps γ: [0, 1] ’ G with γ(0) = e.) Show

˜˜ ˜ ˜

that there is a unique Lie group structure µ: G — G ’ G for which the homotopy class of

˜˜ ˜

the constant map e ∈ G is the identity and so that π is a homomorphism. (Hints: Give G

the (unique) smooth structure for which π is a local di¬eomorphism. The multiplication

µ can then be de¬ned as follows: The map µ = µ —¦ (π — π): G — G ’ G is a smooth map

˜ ˜

˜ ¯

˜ ˜

and satis¬es µ(˜, e) = e. Since G — G is simply connected, the universal lifting property

¯e˜

˜˜ ˜ ˜

of the covering map π implies that there is a unique map µ: G — G ’ G which satis¬es

π —¦ µ = µ and µ(˜, e) = e. Show that µ is smooth, that it satis¬es the axioms for a group

˜¯ ˜e˜ ˜ ˜

multiplication (associativity, existence of an identity, and existence of inverses), and that π

is a homomorphism. You will want to use the universal lifting property of covering spaces

a few times.)

˜

The kernel of π is a discrete normal subgroup of G. Show that this kernel lies in the

center of G. (Hint: For any z ∈ ker(π), the connected set {aza’1 | a ∈ G} must also lie

in ker(π).)

Show that the center of the simply connected Lie group

ab

a ∈ R+ , b ∈ R

G=

01

E.2.3 35

is trivial, so any connected Lie group with the same Lie algebra is actually isomorphic

to G.

(In the next Lecture, we will show that whenever K is a closed normal subgroup of

a Lie group G, the quotient group G/K can be given the structure of a Lie group. Thus,

in many cases, one can e¬ectively list all of the connected Lie groups with a given Lie

algebra.)

20. Show that SL(2, R) is not a matrix group! In fact, show that any homomorphism

φ: SL(2, R) ’ GL(n, R) factors through the projections SL(2, R) ’ SL(2, R). (Hint: Re-

call, from earlier exercises, that the inclusion map SL(2, R) ’ SL(2, C) induces the zero

map on π1 since SL(2, C) is simply connected. Now, any homomorphism φ: SL(2, R) ’

GL(n, R) induces a Lie algebra homomorphism φ (e): sl(2, R) ’ gl(n, R) and this may

clearly be complexi¬ed to yield a Lie algebra homomorphism φ (e)C : sl(2, C) ’ gl(n, C).

Since SL(2, C) is simply connected, there must be a corresponding Lie group homorphism

φC : SL(2, C) ’ GL(n, C). Now suppose that φ does not factor through SL(2, R), i.e.,

that φ is non-trivial on the kernel of SL(2, R) ’ SL(2, R), and show that this leads to a

contradiction.)

21. An ideal in a Lie algebra g is a linear subspace h which satis¬es [h, g] ‚ h. Show that

the kernel k of a Lie algebra homomorphism •: h ’ g is an ideal in h and that the image

•(h) is a subalgebra of g. Conversely, show that if k ‚ h is an ideal, then the quotient

vector space h/k carries a unique Lie algebra structure for which the quotient mapping

h ’ h/k is a homomorphism.

Show that the subspace [g, g] of g which is generated by all brackets of the form [x, y]

is an ideal in g. What can you say about the quotient g/[g, g]?

22. Show that, for a connected Lie group G, a connected Lie subgroup H is normal if and

only if h is an ideal of g. (Hint: Use Proposition 7 and the fact that H ‚ G is normal if

and only if ex He’x = H for all x ∈ g.)

23. For any Lie algebra g, let z(g) ‚ g denote the kernel of the homomorphism ad: g ’

gl(g). Use Theorem 2 and Exercise 16 to prove Theorem 4 for any Lie algebra g for which

z(g) = 0. (Hint: Look at the discussion after the statement of Theorem 4.)

Show also that if g is the Lie algebra of the connected Lie group G, then the connected

Lie subgroup Z(g) ‚ G which corresponds to z(g) lies in the center of G. (In the next

lecture, we will be able to prove that the center of G is a closed Lie subgroup of G and

that Z(g) is actually the identity component of the center of G.)

24. For any Lie algebra g, there is a canonical bilinear pairing κ: g — g ’ R, called the

Killing form, de¬ned by the rule:

κ(x, y) = tr ad(x)ad(y) .

E.2.4 36

(i) Show that κ is symmetric and, if g is the Lie algebra of a Lie group G, then κ is

Ad-invariant:

κ Ad(g)x, Ad(g)y = κ(x, y) = κ(y, x).

Show also that

κ [z, x], y = ’κ x, [z, y] .

A Lie algebra g is said to be semi-simple if κ is a non-degenerate bilinear form on g.

(ii) Show that, of all the 2- and 3-dimensional Lie algebras, only so(3) and sl(2, R) are

semi-simple.

(iii) Show that if h ‚ g is an ideal in a semi-simple Lie algebra g, then the Killing form

of h as an algebra is equal to the restriction of the Killing form of g to h. Show also

that the subspace h⊥ = {x ∈ g | κ(x, y) = 0 for all y ∈ h} is also an ideal in g and that

g = h • h⊥ as Lie algebras. (Hint: For the ¬rst part, examine the e¬ect of ad(x) on a

basis of g chosen so that the ¬rst dim h basis elements are a basis of h.)

(iv) Finally, show that a semi-simple Lie algebra can be written as a direct sum of ideals

hi , each of which has no proper ideals. (Hint: Apply (iii) as many times as you can

¬nd proper ideals of the summands found so far.)

A more general class of Lie algebras are the reductive ones. We say that a Lie algebra

is reductive if there is a non-degenerate symmetric bilinear form ( , ): g — g ’ R which

satisifes the identity [z, x], y + x, [z, y] = 0. Using the above arguments, it is easy to

see that a reductive algebra can be written as the direct sum of an abelian algebra and

some number of simple algebras in a unique way.

25. Show that, if ω is the canonical left-invariant 1-form on G and Yv is the right-invariant

vector ¬eld on G satisfying Yv (e) = v, then

ω Yv (a) = Ad a’1 (v).

(Remark: For any skew-commutative algebra (a, [, ]), the function [[, ]]: a — a — a ’ a

de¬ned by

[[x, y, z]] = [[x, y], z] + [[y, z], x] + [[z, x], y]

is tri-linear and skew-symmetric, and hence represents an element of a — Λ3 (a— ).)

E.2.5 37

Lecture 3:

Group Actions on Manifolds

In this lecture, I turn from the abstract study of Lie groups to their realizations as

“transformation groups.”

Lie group actions.

De¬nition 1: If (G, µ) is a Lie group and M is a smooth manifold, then a left action of

G on M is a smooth mapping »: G — M ’ M which satis¬es »(e, m) = m for all m ∈ M

and

»(µ(a, b), m) = »(a, »(b, m)).

Similarly, a right action of G on M is a smooth mapping ρ: M — G ’ M, which satis¬es

ρ(m, e) = m for all m ∈ M and

ρ(m, µ(a, b)) = ρ(ρ(m, a), b).

For notational sanity, whenever the action (left or right) can be easily inferred from

context, we will usually write a · m instead of »(a, m) or m · a instead of ρ(m, a). Thus,

for example, the axioms for a left action in this abbreviated notation are simply e · m = m

and a · (b · m) = ab · m.

For a given a left action »: G — M ’ M, it is easy to see that for each ¬xed a ∈ G

the map »a : M ’ M de¬ned by »a (m) = »(a, m) is a smooth di¬eomorphism of M onto

itself. Thus, G gets represented as a group of di¬eomorphisms, or “transformations” of a

manifold M. This notion of “transformation group” was what motivated Lie to develop his

theory in the ¬rst place. See the Appendix to this Lecture for a more complete discussion

of this point.

Equivalence of Left and Right Actions. Note that every right action ρ: M — G ’

M can be rewritten as a left action and vice versa. One merely de¬nes

ρ(a, m) = ρ(m, a’1 ).

˜

(The reader should check that this ρ is, in fact, a left action.) Thus, all theorems about

˜

left actions have analogues for right actions. The distinction between the two is mainly

for notational and conceptual convenience. I will concentrate on left actions and only

occasionally point out the places where right actions behave slightly di¬erently (mainly

changes of sign, etc.).

Stabilizers and Orbits. A left action is said to be e¬ective if g · m = m for all

m ∈ M implies that g = e. (Sometimes, the word faithful is used instead.) A left action

is said to be free if g = e implies that g · m = m for all m ∈ M.

A left action is said to be transitive if, for any x, y ∈ M, there exists a g ∈ G so that

g · x = y. In this case, M is usually said to be homogeneous under the given action.

L.3.1 38

For any m ∈ M, the G-orbit of m is de¬ned to be the set

G · m = {g · m | g ∈ G}

and the stabilizer (or isotropy group) of m is de¬ned to be the subset

Gm = {g ∈ G | g · m = m}.

Note that

Gg·m = g Gm g ’1 .

Thus, whenever H ‚ G is the stabilizer of a point of M, then all of the conjugate subgroups

of H are also stabilizers. These results imply that

GM = Gm

m∈M

is a closed normal subgroup of G and consists of those g ∈ G for which g · m = m for all

m ∈ M. Often in practice, GM is a discrete (in fact, usually ¬nite) subgroup of G. When

this is so, we say that the action is almost e¬ective.

The following theorem says that orbits and stabilizers are particularly nice objects.

Though the proof is relatively straightforward, it is a little long, so we will consider a few

examples before attempting it.

Theorem 1: Let »: G — M ’ M be a left action of G on M. Then, for all m ∈ M, the

stabilizer Gm is a closed Lie subgroup of G. Moreover, the orbit G · m can be given the

structure of a smooth submanifold of M in such a way that the map φ: G ’ G · m de¬ned

by φ(g) = »(g, m) is a smooth submersion.