adjoint representation operatorsAd(g) discussed in chapter 5. So, in this case
the corresponding action by automorphisms on the Lie algebragis the adjoint
action
X∈g→φg(X) =Ad(g)(X) =gXg−^1
The infinitesimal version of the Lie group adjoint representation byAd(g)
ongis the Lie algebra adjoint representation by operatorsad(Z) ong
X∈g→Z·X=ad(Z)(X) = [Z,X]This is an action ofgon itself by derivations.
16.2.2 The symplectic group as automorphism group
Recall the definition 13.2 of the Heisenberg groupH 3 as elements
((
x
y
)
,z)
∈R^2 ⊕R
with the group law
((
x
y
)
,z)((
x′
y′)
,z′)
=
((
x+x′
y+y′)
,z+z′+1
2
Ω
((
x
y)
,
(
x′
y′)))
Elementsg∈SL(2,R) act onH 3 by
((
x
y)
,z)
→Φg(((
x
y)
,z))
=
(
g(
x
y)
,z)
(16.18)
HereG=SL(2,R),H=H 3 and Φggiven above is an action by automorphisms
since
Φg((
x
y)
,z)
Φg((
x′
y′)
,z′)
=
(
g(
x
y)
,z)(
g(
x′
y′)
,z′)
=
(
g(
x+x′
y+y′)
,z+z′+1
2
Ω
(
g(
x
y)
,g(
x′
y′)))
=
(
g(
x+x′
y+y′)
,z+z′+1
2
Ω
((
x
y)
,
(
x′
y′)))
=Φg(((
x
y)
,z)((
x′
y′)
,z′))
(16.19)
Recall that, in the exponential coordinates we use, the exponential map
between the Lie algebrah 3 and the Lie groupH 3 is the identity map, with both
h 3 andH 3 identified withR^2 ⊕R. As in section 14.2. we will explicitly identify
h 3 with functionscqq+cpp+conM, writing these as
((
cq
cp)
,c