The commutation relations amongst these matrices are
[E,F] =G [G,E] = 2E [G,F] =− 2 Fwhich are the same as the Poisson bracket relations between the corresponding
quadratic polynomials.
The moment map for theSL(2,R) action onM=R^2 of equation 16.3 is
given by
μL=−aqp+bq^2
2−
cp^2
2(16.9)
To check this, first compute using the definition of the Poisson bracket
−XμLF(q,p) ={μL,F}= (bq−ap)∂F
∂p
+ (aq+cp)∂F
∂qElementsetL∈SL(2,R) act on functions onMby
etL·F(q(m),p(m)) =F(q(e−tL·m),p(e−tL·m))where (form∈Mwritten as column vectors)e−tL·mis multiplication by the
matrixe−tL. On linear functionsl∈ Mwritten as column vectors, the same
group action takesltoetLland acts on basis vectorsq,pofMby
(
q
p
)
→(etL)T(
q
p)
The vector fieldXLis then given by
−XLF(q,p) =
d
dtF((q(e−tL·m),p(e−tL·m))t=0=
(
∂F
∂q,
∂F
∂p)
·
d
dt
(etL)T(
q
p)
t=0
=(
∂F
∂q,
∂F
∂p)
·LT
(
q
p)
=
(
∂F
∂q,
∂F
∂p)
·
(
aq+cp
bq−ap)
and one sees thatXL=XμLas required. The isomorphism of the theorem is the
statement thatμLhas the Lie algebra homomorphism property characterizing
moment maps:
{μL,μL′}=μ[L,L′]
Two important subgroups ofSL(2,R) are
- The subgroup of elements one gets by exponentiatingG, which is isomor-
phic to the multiplicative group of positive real numbers
etG=(
et 0
0 e−t)
Here one can explicitly see that this group has elements going off to infinity.