Lie Groups, Spring 2014 (GeomLie)
EXERCISE CLASSES
Week 1, 5/2: Ex. 2, 5, 6, 8, 9, 11, 12
Week 2, 12/2: Ex. 6, 11, 12 (again). 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?
4) Let G be the set of all matrices g in GL(n+1,R) for which the last row
has the form (0,...,0,1).
Show G is a Lie group and determine its Lie algebra.
For n=1: Determine the exponential map explicitly and show it is surjective onto the identity component of G.
Show the surjectivity statement is not valid if n>1.
5) Define gl(V)β={X in Mn(R)| β(Xu,v)=-β(u,Xv) for all u,v}
in analogy with GL(V)β (page 12).
Show it is a Lie subalgebra of gl(V), and that
it is the Lie algebra of GL(V)β
6) Show O(n)=SO(n) x Z/2Z as a manifold for all n.
Determine for which n
this diffeomorphism can be attained by an isomorphism of Lie groups.
Week 3, 19/2: Hand in (non-mandatory) assignment.
For the classroom:
14 (assume H1 and H2 have countably many
components), 17 (the notions "ideal" and "normal subgroup" are defined
in the preceding exercises), 18, 20 (here "generate" means "span"), 21
Week 4, 26/2: 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.)
Now specialize to G=O(n) and V=Rn with G acting by rotations.
In this case E(n)=GxV is called the Euclidean group (it consists of
all isometries of Rn). Construct a Lie homomorphism of E(n) into the Lie group G of Week2, Extra 4 (see above).
Week 5, 5/3: Hand in mandatory 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 3).
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?