Lie Groups, Spring 2015 (GeomLie)

See also (requires login) Absalon.

LECTURES

There are lectures by H. Schlichtkrull every Monday, 14:15-16:00 in Aud 10 (HC), and Wednesday, 10:15-12:00 in Aud 06 (HC). Lecture notes by Erik van den Ban can be downloaded from here. See also the exercises. The author's own lectures on the notes have been recorded, see here (you can also find them in youtube).

Week 1, Monday 2/2: Chapters 1-3
Week 1, Wednesday 4/2: Chapter 3 and the extra notes on flows.

Week 2, Monday 9/2: Chapters 4-5 (See the extra
notes on the adjoint representation and the Lie bracket.)
Week 2, Wednesday 11/2: Chapter 7 (we will read Chapters 6-7 in opposite order) (See the extra
notes on good, bad and ugly Lie subgroups.)

Week 3, Monday 16/2: Chapters 6 and 8
Week 3, Wednesday 18/2: Chapters 8-9 (See the extra details for the proof of
Lemma 8.2.)

Week 4, Monday 23/2: Cor 9.3 + Chapters 10-11
Week 4, Wednesday 25/2: Chapter 12

Week 5, Monday 2/3: Chapters 13-14
Week 5, Wednesday 4/3: Chapters 15-17 (Chapter 15: no details. Chapter 18 will be skipped)

Week 6, Monday 9/3: Chapter 19 (See the extra
notes on Borel measures.)
Week 6, Wednesday 11/3: Chapter 20 (Chapters 21-28 will be skipped). This lecture will be given by Tyrone.

Week 7, Monday 16/3: Chapters 29,30.
Week 7, Wednesday 18/3: Chapter 30,31.

Week 8, Monday 23/3: Chapters 31-32.
Week 8, Wednesday 25/3: Chapter 33-35.

Week 9, Wednesday 8/4: Chapter 36 (skip 37). Overview of 38.

EXERCISE CLASSES and ASSIGNMENTS

There are exercise classes, supervised by
Tyrone Crisp, every Wednesday, 13:15-16:00 in 1-0-26 (in the building of DIKU).
Most of the exercises can be found
here.

Week 1, 4/2: Ex. 2, 5, 6, 8, 9, 11, 12

Week 2, 11/2: Ex. 6, 11, 12 (if they were not done). Extra:
1) With exercise 9(a), determine a maximal neighborhood U of 0 in so(2) for which φ→exp(φ): U→SO(2) is injective.
With exercise 9(c), show the X is not uniquely determined by x.
2) With exercise 11(a), show the statement is valid also for the matrix obtained from y by replacing the element in the upper right with an arbitrary non-zero number. Determine the set of all upper triangular matrices which belong to the exponential image.
Show also that exp is not surjective onto SL(n,R) for n>2.
3) With exercise 11(b), show Xs is uniquely determined by x. Is this the case for Xa?

Week 3, 18/2: Hand in (non-mandatory) assignment. For the classroom:
17 (the notions "ideal" and "normal subgroup" are defined in the preceding exercises), 18, 20 (here "generate" means "span"), 21. Extra:
In continuation of the hand-in: Determine the Lie algebras of K and B, and show that together they span the Lie algebra of G. Show that the map K× B→ G is a local diffeomorphism at (e,e). Conclude finally that this map is a diffeomorphism.

Week 4, 25/2: Exercise 19. Extra:
1) Let H be a one-parameter subgroup of a Lie group G, and assume H is not closed. Show that then the closure of H is compact.
2) Let H be a discrete normal subgroup of a connected Lie group G. Show H is contained in the center of G.
3) Let G be a Hausdorff topological group. Let A be a closed subset and B a compact subset. Show that the set AB of all products ab of elements from A and B is closed.
Now let H be a subgroup of G and equip G/H with the quotient topology. Show that G/H is Hausdorff if and only if H is closed in G.
4) Let G be a Hausdorff topological group and H a closed subgroup. Equip G/H with the quotient topology. Show that if H and G/H are connected, then so is G.
5) Let G=SO(n) and H=SO(n-1) where n>1, and embed H in the lower right corner of G. Show that G/H with the quotient topology is homeomorphic to the unit sphere Sn-1 in Rn, the G-orbit through e1. Assuming it to be known that Sn is connected for all n>1, give a quick proof that SO(n) is connected for all n. Where does the same proof fail for O(n)? Now generalize to SU(n) and U(n).
6) Let G be a Lie group and V a finite-dimensional vector space, on which G acts from the left by linear endomorphisms. Write the action of g on a vector v as gv. Equip the Cartesian product GxV with the structure of the product manifold, and with the mulitplication defined by
(g1,v1).(g2,v2)=(g1g2, v1+g1v2)
Show that GxV is a Lie group and that Gx{0} and {e}xV are closed Lie subgroups isomorphic to G and V. Show also that V is a normal subgroup of GxV. (GxV is said to be the semidirect product of G and V.)
In the special case G=O(n) and V=Rn with G acting by rotations, E(n)=GxV is called the Euclidean group (it consists of all isometries of Rn).

Week 5, 4/3: Mandatory hand in
assignment. For the classroom:
1) Let G and H be connected Lie groups. Show that every discrete normal subgroup of G is contained in the center of G. Let f: G→ H be a homomorphism of Lie groups. Show that if f has injective differential f*, then its kernel is central in G. Show also that if f has surjective differential f*, then f is surjective.
2) Consider the circle subgroup T of SU(2) defined on page 42. Show by means of Proposition 10.3 that SU(2)/T is diffeomorphic to a 2-sphere S2. Show that this provides a fibration of S3 as a S1 bundle over S2 (the Hopf fibration).
3) Let G be a Lie group and consider the diagonal subgroup H in GxG of all elements of the form (g,g). Verify it is a closed Lie subgroup. Show that by g → (g,e)H one obtains a diffeomorphism of G onto GxG/H.
4) Show that every continuous action of a compact Hausdorff group H on a locally compact Hausdorff space M is proper. Is it also true that every continuous action of a locally compact Hausdorff group on a compact Hausdorff space is proper?
5) Let G be a locally compact Hausdorff group and H a subgroup with the inherited topology. Show that if the right action of H on G is proper then H is closed (the converse to Lemma 14.1).
6) Give an example of a free smooth action which is not proper and an example of a proper smooth action which is not free.
7) Verify that the following is a smooth right action of H=R on R2: (x,y).t=(x+ty,y). Determine the orbit space and its topology. Find a compact set C in R2 for which CH is not closed, and conclude the action is not proper. Is it free?
8) Consider a continuous action of a group H on M, both assumed to be locally compact Haudorff. Show that (m,h)→mh is a proper map from MxH to M if and only if H is compact (in which case the action is proper according to Exercise 4).
9) Show that the map φ from H=R2 to SL(3,R), which maps (x,y) to the 3x3 matrix with rows (1,x,y),(0,1,0),(0,0,1) respectively, is a homomorphism. Verify that a right action of H is defined on the projective space M=RP2 by [v].h=[vφ(h)], where v in R3 is a row vector. Show there is an open dense orbit, and that its complement is a line consisting of H-fixed points. What is then the topology of the orbit space?

Week 6, 11/3: See
here (only exercises 1-8).

Week 7, 18/3: Hand in mandatory assignment. For the classroom:
Exercise 24 (real numbers only), 20.24 (page 75), 26, 27, 28, prove Lemma 29.3.
Extra: Define Wn to be the n+1 dimensional vector space of polynomials p in one complex variable z, of degree less than or equal to n. Define for g in SU(2) with coordinates α and β as in Example 20.25, and p in Wn:
[ρ(g)p](z)=(-βz+α)n p((α*z+β*)/(-βz+α))
where the asterisk denotes complex conjugation. Show that (ρ,Wn) is a representation of SU(2). Show that it is equivalent to (π,Pn) from the example, by finding an explicit intertwining isomorphism.

Week 8, 25/3:
1) Consider the Lie algebra so(n) of SO(n), which consists of the skew-symmetric nxn matrices.
Show that its complexification can be identified with the space of all nxn complex matrices which are skew symmetric.
If n=2k or n=2k+1, find a k-dimensional torus t in so(n), and show it is maximal.
Determine the weight space decomposition (see lemma 31.5) of the standard representation of SO(n) on Cn.
2) Determine for n=3 vectors H,X and Y in so(3) with the same commutation relations as in Example 31.8, and find the root space decomposition of so(3,C).
Determine also the root space decomposition of so(4,C).
3) Let G be a Lie group and let R be a subset of the Lie algebra of G. The centralizer ZG(R) of R in G is defined to be the set of elements g for which Ad(g)X=X for all X in R.
Show that this is a closed Lie subgroup of G, and that its Lie algebra is the space of all elements in the Lie algebra of G which commute with every element from R.
4) Let G be a compact Lie group and let t be a maximal torus in its Lie algebra.
Show that T=exp(t) is a closed Lie subgroup of G.
Hint: Use the previous exercise with R=t. Conclude that the image of the Lie algebra of G by the exponential map is closed in G.
It can be shown that the image is also open (but this is more difficult), and hence equal to G if G is connected.
Thus exp is surjective for every compact connected Lie group.
5) Let G be a compact connected Lie group and let T=exp(t) where t is a maximal torus.
Let N=NG(t) denote the normalizer of t in G, that is, the set of elements g for which Ad(g)(t)=t.
Show that this is a closed Lie subgroup of G, and that its Lie algebra is the space of all elements X in the Lie algebra of G, such that ad(X)(t)=t (the normalizer of t in the Lie algebra).
Show that an element X in the Lie algebra of G, which normalizes t as above, necessarily belongs to t (use Corollary 31.7).
Conclude that NG(t) has Lie algebra t, and that NG(t)/T is a finite group.
It can be shown that this group is isomorphic to the Weyl group in Definition 36.8

Week 9, 8/4: There is only one exercise: The root system of SO(7) In addition you should agree with Tyrone on how to hand in the: Last assignment April 13 assignment.

In week 3 a (non-mandatory) assignment can be handed in at the start of the exercise class. In week 5, 7 and 9 of the course mandatory assignments are to be handed in at the start of the exercise class. These assignments are given one week in advance and will be graded. The final grade for the course will be based on the grades for the mandatory assignments: the first two with weight 0.25, the last with weight 0.5.