Homework 2—Differential Geometry
Due date: 20.4. 2021
1. Show that the cross-product × : R3 × R3 → R3 defines a Lie bracket on R3 and that the
Lie algebra (R3, ×) is isomorphic to the Lie algebra (so(3, R), [·, ·]).
2. Consider the Lie group SL(2, K) and its Lie algebra sl(3, K) for K = R or C.
• Show that ·, · : sl(2, K) × sl(2, K) → K defined by
X, Y =
1
2
trace(XY )
defines a symmetric non-degenerate K-bilinear form on the 3-dimensional K-vector
space sl(2, K). Moreover, show that over R it has signature (2, 1).
• Show that the adjoint representation Ad : SL(2, K) → GL(sl(2, K)) ∼= GL(3, K)
induces covering maps
Ad : SL(2, C) → SO((sl(2, C), ·, · ) ∼= SO(3, C)
Ad : SL(2, R) → SO((sl(2, R), ·, · ) ∼= SOo(2, 1).
What are the kernels of these group homomorphisms?
3. Consider the upper-half plane H = {(x, y) ∈ R2 : y > 0} = {z ∈ C : Im(z) > 0}.
• Show that
SL(2, R) × H → H
(
a b
c d
, z) →
az + b
cz + d
defines a smooth transitive left action of SL(2, R) on H. What is the isotropy group of
i ∈ H ⊂ C?
• Consider the following Riemannian metric on H:
g =
dx2 + dy2
y2
=
4|dz|2
|z − ¯z|2
(z = x + iy, dz = dx + idy).
Show that SL(2, R) acts by isometries on (H, g), that is, z → az+b
cz+d is an isometry for
any
a b
c d
∈ SL(2, R).
4. Suppose G is a connected Lie group. Let φ : G → GL(V ) be a representation on a finitedimensional
vector space V and let φ : g → gl(V ) be the induced representation of the Lie
algebra g of G.
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• Show that a subspace W ⊂ V is G-invariant ⇐⇒ it is g-invariant.
• Show that V is unitary as G-representation ⇐⇒ it is unitary as g-representation.
5. Suppose G is a compact Lie group of dimension n. Choose a nonzero element ω ∈ Λng∗
(i.e. a volume form on the vector space g). Then via left-multiplication this gives rise to a
volume form on G:
vol(g)(ξ1, ...ξn) = ω(Tgλg−1 ξ1, ...., Tgλg−1 ξn), for ξ1, ...., ξn ∈ TgG.
Hence, we can integrate smooth functions f : G → K = R, C by setting
G
f :=
G
fvol.
• Show that vol is left-invariant (i.e. λ∗
gvol = vol for all g ∈ G) and deduce that G f =
G f ◦ λg for all g ∈ G.
• Let V be a real or complex representation of G and let b(·, ·) : V × V → K be an
arbitrary positive definite (Hermitian in the complex case) inner product on V . For two
vectors v, w ∈ V set
v, w :=
G
fv,w,
where fv.w : G → K is the smooth function defined by fv.w(g) = b(g−1v, g−1w).
Show that ·, · defines a G-invariant positive definite inner product on V (Hermitian
in the complex case), i.e. V is a unitary representation.