218
V. K. Dobrev
applied to the Schrödinger algebra. This is a project we start in the present short
paper. Actually here we give as examples the low level singular vectors of Verma
modules over the Jacobi algebra.
2 Preliminaries
The Jacobi algebra is the semi-direct sum G n := H n sp(n, R) C [4, 5]. The
Heisenberg algebra H n is generated by the boson creation (respectively, annihilation) operators a
+
i (a
−
i ), i, j = 1, . . . , n, which verify the canonical commutation
relations
a
−
i , a
+
j
= δ ij ,
[a
−
i , a
−
j ] =
a
+
i , a
+
j
= 0.
(1)
H n is an ideal in G n , i.e., [H n , G n ] = H n , determined by the commutation relations
(following the notation of [6]):
a
+
k , K
+
ij
= [a
−
k , K
−
ij ] = 0,
(2a)
[a
−
i , K
+
kj ] =
1
2 δ ik a
+
j +
1
2 δ ij a
+
k ,
K
−
kj , a
+
i
=
1
2 δ ik a
−
j +
1
2 δ ij a
−
k , (2b)
K
0
ij , a
+
k
=
1
2 δ jk a
+
i ,
a
−
k , K
0
ij
=
1
2 δ ik a
−
j .
(2c)
K
±,0
ij are the generators of the S n ≡ sp(n, R) C algebra:
[K
−
ij , K
−
kl ] = [K
+
ij , K
+
kl ] = 0,
2
K
−
ij , K
0
kl
= K
−
il δ kj + K
−
jl δ ki ,
(3a)
2[K
−
ij , K
+
kl ] = K
0
kj δ li + K
0
lj δ ki + K
0
ki δ lj + K
0
li δ kj
(3b)
2
K
+
ij , K
0
kl
= −K
+
ik δ jl − K
+
jk δ li , 2
K
0
ji , K
0
kl
= K
0
jl δ ki − K
0
ki δ lj . (3c)
In order to implement our approach we introduce a triangular decomposition
of G n :
G n = G
+
n ⊕ K n ⊕ G
−
n ,
(4)
using the triangular decomposition S n = S +
n ⊕ K n ⊕ S −
n , where:
G
±
n = H
±
n ⊕ S
±
n
(5)
H
±
n = l.s.{ a
±
i : i = 1, . . . , n} ,
S
+
n = l.s.{ K
+
ij : 1 ≤ i ≤ j ≤ n} ⊕ l.s.{ K
0
ij : 1 ≤ i < j ≤ n}
S
−
n = l.s.{ K
−
ij : 1 ≤ i ≤ j ≤ n} ⊕ l.s.{ K
0
ij : 1 ≤ j < i ≤ n}
K n = l.s.{ K
0
ii : 1 ≤ i ≤ n}.