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