wherek∈Sp(2d,R), then the automorphism Φkthat defines the Jacobi group
is given by the one studied in section 16.2
Φk(((
cq
cp)
,c))
=
(
k(
cq
cp)
,c)
(18.1)
Note that the Euclidean groupE(d) is a subgroup of the Jacobi groupGJ(d),
the subgroup of elements of the form
(((
0
cp
)
, 0
)
,
(
R 0
0 R
))
whereR∈SO(d). The ((
0
cp
)
, 0
)
⊂H 2 d+1make up the groupRdof translations in theqjcoordinates, and the
k=(
R 0
0 R
)
⊂Sp(2d,R)are symplectic transformations since
Ω
(
k(
cq
cp)
,k(
c′q
c′p))
=Rcq·Rc′p−Rcp·Rc′q=cq·c′p−cp·c′q=Ω((
cq
cp)
,
(
c′q
c′p))
(Ris orthogonal so preserves dot products).
18.3 Semi-direct product Lie algebras
We have seen that semi-direct product Lie groups can be constructed by taking
a productN×Kof Lie groups as a set, and imposing a group multiplication
law that uses an action ofK onNby automorphisms. In a similar manner,
semi-direct product Lie algebrasnokcan be constructed by taking the direct
sum ofnandkas vector spaces, and defining a Lie bracket that uses an action of
konnby derivations (the infinitesimal version of automorphisms, see equation
16.17).
Considering first the exampleE(d) =RdoSO(d), recall that elementsE(d)
can be written in the form (
R a
0 1
)
forR∈SO(d) anda∈Rd. The tangent space to this group at the identity will
be given by matrices of the form
(
X a
0 0