Theorem 16.3.Thesp(2d,R)action onh 2 d+1=M⊕Rby derivations is
L·(cq·q+cp·p+c) ={μL,cq·q+cp·p+c}=c′q·q+c′p·p (16.22)
where (
c′q
c′p
)
=L
(
cq
cp
)
or, equivalently (see section 4.1), on basis vectors ofMone has
{
μL,
(
q
p
)}
=LT
(
q
p
)
Proof.One can first prove 16.22 for the cases when only one ofA,B,Cis non-
zero, then the general case follows by linearity. For instance, taking the special
case
L=
(
0 B
0 0
)
, μL=
1
2
q·Bq
the action on coordinate functions (the basis vectors ofM) is
{
1
2
q·Bq,
(
q
p
)
}=LT
(
q
p
)
=
(
0
Bq
)
since
{
1
2
∑
j,k
qjBjkqk,pl}=
1
2
∑
j,k
(qj{Bjkqk,pl}+{qjBjk,pl}qk)
=
1
2
(
∑
j
qjBjl+
∑
k
Blkqk)
=
∑
j
Bljqj (sinceB=BT)
Repeating forAandCgives in general
{
μL,
(
q
p
)}
=LT
(
q
p
)
We can now prove theorem 16.2 as follows:
Proof.
L→μL
is clearly a vector space isomorphism of matrices and of quadratic polynomials.
To show that it is a Lie algebra isomorphism, the Jacobi identity for the Poisson
bracket can be used to show
{μL,{μL′,cq·q+cp·p}}−{μL′,{μL,cq·q+cp·p}}={{μL,μL′},cq·q+cp·p}