Definition(Heisenberg Lie algebra).The Heisenberg Lie algebrah 2 d+1is the
vector spaceR^2 d+1=R^2 d⊕Rwith the Lie bracket defined by its values on a
basisXj,Yj,Z(j= 1,...d)by
[Xj,Yk] =δjkZ, [Xj,Z] = [Yj,Z] = 0Writing a general element as
∑d
j=1xjXj+∑d
k=1ykYk+zZ, in terms of coor-
dinates the Lie bracket is
[((
x
y
)
,z)
,
((
x′
y′)
,z)]
=
((
0
0
)
,x·y′−y·x′)
(13.1)
This Lie algebra can be written as a Lie algebra of matrices for anyd. For
instance, in the physical case ofd= 3, elements of the Heisenberg Lie algebra
can be written
0 x 1 x 2 x 3 z
0 0 0 0 y 1
0 0 0 0 y 2
0 0 0 0 y 3
0 0 0 0 0
13.2 The Heisenberg group
Exponentiating matrices inh 3 gives
exp
0 x z
0 0 y
0 0 0
=
1 x z+^12 xy
0 1 y
0 0 1
so the group with Lie algebrah 3 will be the group of upper triangular 3 by 3 real
matrices with 1 on the diagonal, and this group will be the Heisenberg group
H 3. For our purposes though, it is better to work in exponential coordinates
(i.e., labeling a group element with the Lie algebra element that exponentiates
to it). In these coordinates the exponential map relating the Heisenberg Lie
algebrah 2 d+1and the Heisenberg Lie groupH 2 d+1is just the identity map, and
we will use the same notation
((
x
y
)
,z)
for both Lie algebra and corresponding Lie group elements.
Matrix exponentials in general satisfy the Baker-Campbell-Hausdorff for-
mula, which says
eAeB=eA+B+(^12) [A,B]+ 121 [A,[A,B]]− 121 [B,[A,B]]+···
where the higher terms can all be expressed as repeated commutators. This
provides one way of showing that the Lie group structure is determined (for