## Lie Groups, Spring 2014 (GeomLie)

See also (requires login)
Absalon.

### LECTURES

There are lectures by H. Schlichtkrull every Monday, 13:15-15:00 in room A 104 (HCØ), and Wednesday, 13:15-15:00 in Aud 07 (HCØ). 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 3/2: Chapters 1-2
(See the extra notes on flows.)

**Week 1**, Wednesday 5/2: Chapter 3

**Week 2**, Monday 10/2: Chapters 4-5
(See the extra notes on the adjoint representation and the Lie bracket.)

**Week 2**, Wednesday 12/2: Chapter 6-7
(See the extra notes on good, bad and ugly Lie subgroups.)

**Week 3**, Monday 17/2: Chapters 8

**Week 3**, Wednesday 19/2: Chapter 8-9

**Week 4**, Monday 24/2: Chapter 10-11

**Week 4**, Wednesday 26/2: Chapter 12
NB: **Room change: 1-0-04**

**Week 5**, Monday 3/3: Chapters 13-14 (Chapter 15 skipped)

**Week 5**, Wednesday 5/3: Chapters 16-17 (Chapter 18 skipped)

**Week 6**, Monday 10/3: Chapter 19
(See the extra notes on Borel measures.)
(See also this note about the defintion of the modular function.)

**Week 6**, Wednesday 13/3: Chapter 20: Only partial, the representations will be finite dimensional. (Chapters 21-28 skipped)

**Week 7**, Monday 17/3: Chapters 29,30. This lecture will be given by Hans Plesner Jakobsen

**Week 7**, Wednesday 19/3: Chapter 31. This lecture will be given by Hans Plesner Jakobsen

**Week 8**, Monday 24/3: Chapters 32. This lecture will be given by Hans Plesner Jakobsen

**Week 8**, Wednesday 26/3: Chapter 33-35. This lecture will be given by Hans Plesner Jakobsen

**Week 9**, Monday 31/3: Chapter 36 (Chapter 37 skipped)

**Week 9**, Wednesday 2/4: Chapter 38

### EXERCISE CLASSES and ASSIGNMENTS

There are exercise classes, supervised by
Dieter Degrijse, every Wednesday, 9:15-12:00 in 1-0-26 (in the building of DIKU).

Most of the exercises can be found here.

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.

**Assignments:**

**Week 3**, 19/2: Non-mandatory assignment.

**Week 5**, 5/3: Mandatory assignment.

**Week 7**, 19/3: Mandatory assignment.

**Week 9**, 2/4: Hand in mandatory assignment (will be posted in Absalon)

**Classroom:**

The exercises of weeks 1-5 have been moved to here

**Week 6**, 12/3: See here

**Week 7**, 19/3: Hand in mandatory assignment assignment.

For the classroom: Exercise 24 (real numbers only), 20.24 (page 75), 26, 27, 28, prove Lemma 29.3.

*Extra:* Define W_{n} 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 W_{n}:

[ρ(g)p](z)=(-βz+α)^{n}
p((α^{*}z+β^{*})/(-βz+α))

where the asterisk denotes complex conjugation.

Show that (ρ,W_{n}) is a representation
of SU(2).

Show that it is equivalent to (π,P_{n}) from the example, by finding an
explicit intertwining
isomorphism.

**Week 8**, 26/3:

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 C^{n}.

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
Z_{G}(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 (this implies the first statement
in Proposition 37.3).

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 (compare also the second statement in Proposition 37.3).

5) Let G be a compact connected Lie group and let T=exp(t) where t is
a maximal torus.

Let N=N_{G}(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 N_{G}(t)
has Lie algebra t, and that N_{G}(t)/T is a finite group.

It can be shown that this group is isomorphic to the Weyl group in Definition 36.8

6) Exercises 37 (read also 8 lines of text above the exercise) and 38.

**Week 9**, 2/4: Meet at 11:00 and hand in the last mandatory assignment. There is only one exercise:
The root system of SO(7)