Most of the exercises can be found here.

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 X

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.

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 S

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

(g

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

In the special case G=O(n) and V=R

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

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 S

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 R

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=R

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

Extra: Define W

[ρ(g)p](z)=(-βz+α)

where the asterisk denotes complex conjugation. Show that (ρ,W

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

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

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=N

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

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

In week 3 a (non-mandatory) assignment can be handed in at the start of the exercise class.