Homework 1—Differential Geometry
Due date:16.3. 2021
1. Let K = R or C, set
Ip,q :=
Idp 0
0 −Idq
∈ Mp+q(K) Jn :=
0 Idn
−Idn 0
∈ M2n(K),
and consider the following subgroups of the general linear group GL(n, K) (resp. GL(2n, K)).
• The special linear group given by
SL(n, K) = {A ∈ GL(n, K) : detK(A) = 1}.
• The orthogonal and the special orthogonal group
O(n, K) = {A ∈ GL(n, K) : At
= A−1
} and SO(n, K) = O(n, K) ∩ SL(n, K).
Note that A ∈ O(n, K) implies detK(A) = ±1.
• The (indefinite) orthogonal group of signature (p, q) with p + q = n:
O(p, q) = {A ∈ GL(n, R) : At
Ip,qA = Ip,q}.
• The (indefinite) special orthogonal group of signature (p, q) with p + q = n:
SO(p, q) = O(p, q) ∩ SL(n, R).
• The symplectic group
Sp(2n, K) = {A ∈ GL(2n, K) : At
JnA = Jn}.
• The (indefinite) unitary group of signature (p, q) with p + q = n
U(p, q) = {A ∈ GL(n, C) : ¯At
Ip,qA = Ip,q}.
Note that A ∈ U(p, q) implies | detC(A)|2 = 1. Here, ¯A denotes the conjugate of A.
• The (indefinite) special unitary group of signature (p, q) with p + q = n:
SU(p, q) = U(p, q) ∩ SL(n, C).
For q = 0, one also writes U(n) := U(n, 0) = {A ∈ GL(n, C) : ¯At = A−1} and
SU(n) := SU(n, 0).
Show that these groups are Lie groups, compute their dimensions and their Lie algebras
sl(n, K), o(n, K) = so(n, K), o(p, q) = so(p, q), u(p, q) and su(p, q).
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2. Suppose (G, µ, ν, e) is a Lie group with Lie algebra (g, [·, ·]). For X ∈ g denote by LX and
RX the left- respectively right-invariant vector field on G generated by X. Show that the
following holds:
(a) RX = ν∗L−X for all X ∈ g
(b) [RX, RY ] = −R[X,Y ] for all X, Y ∈ g;
(c) [LX, RY ] = 0 for all X, Y ∈ g.
As a hint for (a) note that ν ◦ ρg = λg−1 ◦ ν and for (c) it might help to show that the vector
fields (0, LX) and (RY , 0) on G × G are µ-related to LX and RY respectively.