Quantum Mechanics for Mathematicians

whereXis an antisymmetricdbydmatrix anda∈Rd. Exponentiating such
matrices will give elements ofE(d).
The Lie bracket is then given by the matrix commutator
[(
X 1 a 1
0 0

)

,

(

X 2 a 2

0 0

)]

=

(

[X 1 ,X 2 ] X 1 a 2 −X 2 a 1

0 0

)

(18.2)

We see that the Lie algebra ofE(d) will be given by taking the sum ofRd(the
Lie algebra ofRd) andso(d), with elements pairs (a,X) witha∈RdandXan
antisymmetricdbydmatrix. The infinitesimal version of the rotation action
ofSO(d) onRdby automorphisms

ΦR(a) =Ra

is
d
dt

ΦetX(a)|t=0=

d

dt

(etXa)|t=0=Xa

Just in terms of such pairs, the Lie bracket can be written

[(a 1 ,X 1 ),(a 2 ,X 2 )] = (X 1 a 2 −X 2 a 1 ,[X 1 ,X 2 ])

We can define in general:

Definition(Semi-direct product Lie algebra).Given Lie algebraskandn, and
an action of elementsY∈konnby derivations

X∈n→Y·X∈n

the semi-direct productnokis the set of pairs(X,Y)∈n⊕kwith the Lie bracket

[(X 1 ,Y 1 ),(X 2 ,Y 2 )] = ([X 1 ,X 2 ] +Y 1 ·X 2 −Y 2 ·X 1 ,[Y 1 ,Y 2 ])

One can easily see that in the special case of the Lie algebra ofE(d) this agrees
with the construction above.
In section 16.1.2 we studied the Lie algebra of all polynomials of degree
at most two inddimensional phase space coordinatesqj,pj, with the Poisson
bracket as Lie bracket. There we found two Lie subalgebras, the degree zero
and one polynomials (isomorphic toh 2 d+1), and the homogeneous degree two
polynomials (isomorphic tosp(2d,R)) with the second subalgebra acting on the
first by derivations as in equation 16.22.
Recall from chapter 16 that elements of this Lie algebra can also be written
as pairs (((
cq
cp

)

,c

)

,L

)

of elements inh 2 d+1andsp(2d,R), with this pair corresponding to the polyno-
mial
μL+cq·q+cp·p+c