In terms of such pairs, the Lie bracket is given by
[(((
cq
cp
)
,c)
,L
)
,
(((
c′q
c′p)
,c)
,L′
)]
=
((
L
(
c′q
c′p)
−L′
(
cq
cp)
,Ω
((
cq
cp)
,
(
c′q
c′p)))
,[L,L′]
)
which satisfies the definition above and defines the semi-direct product Lie al-
gebra
gJ(d) =h 2 d+1osp(2d,R)
The fact that this is the Lie algebra of the semi-direct product group
GJ(d) =H 2 d+1oSp(2d,R)follows from the discussion in section 16.2.
The Lie algebra ofE(d) will be a sub-Lie algebra ofgJ(d), consisting of
elements of the form (((
0
cp
)
, 0
)
,
(
X 0
0 X
))
whereXis an antisymmetricdbydmatrix.
Digression.Just asE(d)can be identified with a group ofd+1byd+1matrices,
the Jacobi groupGJ(d)is also a matrix group and one can in principle work with
it and its Lie algebra using usual matrix methods. The construction is slightly
complicated and represents elements ofGJ(d)as matrices inSp(2d+ 1,R). See
section 8.5 of [9] for details of thed= 1case.
18.4 For further reading
Semi-direct products are not commonly covered in detail in either physics or
mathematics textbooks, with the exception of the case of the Poincar ́e group of
special relativity, which will be discussed in chapter 42. Some textbooks that
do cover the subject include section 3.8 of [84], chapter 6 of [39] and [9].