232
J. Gaboriaud et al.
L i−1,i L
2
i,i+1 − (q
1/2
+ q
−1/2 )L i,i+1 L i−1,i L i,i+1 + L
2
i,i+1 L i−1,i = −L i−1,i ,
L i,i+1 L
2
i−1,i − (q
1/2
+ q
−1/2 )L i−1,i L i,i+1 L i−1,i + L
2
i−1,i L i,i+1 = −L i,i+1 ,
[L i,i+1 , L j,j +1 ] = 0 for |i − j | > 1.
We shall use the notation L
±
ik = [L
±
ij , L
±
jk ] q ±1/4 for any i < j < k, and by definition
L
±
i,i+1 = L i,i+1 .
The reductive pair (o q 1/2 (6), U q (su(1, 1)) is the one which is of relevance for
the Askey–Wilson algebra. Let us indicate how o q 1/2 (2n) and U q (su(1, 1)) are
dually represented on the standard state space H of 2n independent q-oscillators
described by operators {A
±
i , A 0
i } such that [A 0
i , A
±
i ] = ±A
±
i , [A
−
i , A
+
i ] = q
A 0
i ,
A
−
i A
+
i − qA
+
i A
−
i = 1, i = 1, . . . , 2n. The algebra U q (su(1, 1)) is represented on
H by using the coproduct to embed it in the tensor product of 2n copies of the
q-deformation of the metaplectic representation, this gives
J
(2n)
0
= Δ
(2n−1)
1
2
A
0
i +
1
2
=
1
2
2n
i=1
A
0
i +
1
2
,
J
(2n)
±
= Δ
(2n−1)
1
[2] q 1/2
(A
±
i )
2
=
1
[2] q 1/2
2n
i=1
(A
±
i )
2
2n
j =i+1
q
A 0
j +
1
2
.
(9)
The algebra o q 1/2 (2n) can also be realized in terms of 2n q-oscillators. The 2n − 1
generators take the form
L i,i+1 = q
−
1
2 (A 0
i +
1
2 )
q
1
4 A
+
i A
−
i+1 − q
−
1
4 A
−
i A
+
i+1
,
i = 1, . . . , 2n − 1.
It can be checked that [J
(2n)
0
, L i,i+1 ] = [J
(2n)
± , L i,i+1 ] = 0, i = 1, . . . , 2n − 1, in
other words, that U q (su(1, 1)) and o q 1/2 (2n) have commuting actions on the Hilbert
space H of 2n q-oscillators. This sets the stage for Howe duality. In order to connect
with the Askey–Wilson algebra we take n = 3. The expressions of the operators K A
and K B acting on H that realize the relations (8) (together with the specific central
elements) are rather involved and we shall refer the reader to [10] for the formulas.
We shall only stress that these operators can be obtained in a dual way: They are
affinely related to the generators of the commutant of o q 1/2 (2) ⊕3 in o q 1/2 (6) as well
as to the intermediate U q (su(1, 1)) Casimir elements C (1234) = Δ (3) (C)⊗1⊗1 and
C (3456) = 1 ⊗ 1 ⊗ Δ (3) (C) of the q-metaplectic representation (see (9)). This can be
extended to higher ranks by letting n be arbitrary [11]. For n = 2 we are looking at
the Clebsch–Gordan problem for U q (su(1, 1)). The q-Hahn algebra that arises has
two dual realizations [12]: one as the commutant of o q 1/2 (2) ⊕2 in U q (u(4)) and the
other in terms of the following two U q (su(1, 1)) operators, (Δ(J 0 )⊗1⊗1)−(1⊗1⊗
Δ(J 0 )) and Δ (2) (C) (the full Casimir element) in the q-metaplectic representation.
J. Gaboriaud et al.
L i−1,i L
2
i,i+1 − (q
1/2
+ q
−1/2 )L i,i+1 L i−1,i L i,i+1 + L
2
i,i+1 L i−1,i = −L i−1,i ,
L i,i+1 L
2
i−1,i − (q
1/2
+ q
−1/2 )L i−1,i L i,i+1 L i−1,i + L
2
i−1,i L i,i+1 = −L i,i+1 ,
[L i,i+1 , L j,j +1 ] = 0 for |i − j | > 1.
We shall use the notation L
±
ik = [L
±
ij , L
±
jk ] q ±1/4 for any i < j < k, and by definition
L
±
i,i+1 = L i,i+1 .
The reductive pair (o q 1/2 (6), U q (su(1, 1)) is the one which is of relevance for
the Askey–Wilson algebra. Let us indicate how o q 1/2 (2n) and U q (su(1, 1)) are
dually represented on the standard state space H of 2n independent q-oscillators
described by operators {A
±
i , A 0
i } such that [A 0
i , A
±
i ] = ±A
±
i , [A
−
i , A
+
i ] = q
A 0
i ,
A
−
i A
+
i − qA
+
i A
−
i = 1, i = 1, . . . , 2n. The algebra U q (su(1, 1)) is represented on
H by using the coproduct to embed it in the tensor product of 2n copies of the
q-deformation of the metaplectic representation, this gives
J
(2n)
0
= Δ
(2n−1)
1
2
A
0
i +
1
2
=
1
2
2n
i=1
A
0
i +
1
2
,
J
(2n)
±
= Δ
(2n−1)
1
[2] q 1/2
(A
±
i )
2
=
1
[2] q 1/2
2n
i=1
(A
±
i )
2
2n
j =i+1
q
A 0
j +
1
2
.
(9)
The algebra o q 1/2 (2n) can also be realized in terms of 2n q-oscillators. The 2n − 1
generators take the form
L i,i+1 = q
−
1
2 (A 0
i +
1
2 )
q
1
4 A
+
i A
−
i+1 − q
−
1
4 A
−
i A
+
i+1
,
i = 1, . . . , 2n − 1.
It can be checked that [J
(2n)
0
, L i,i+1 ] = [J
(2n)
± , L i,i+1 ] = 0, i = 1, . . . , 2n − 1, in
other words, that U q (su(1, 1)) and o q 1/2 (2n) have commuting actions on the Hilbert
space H of 2n q-oscillators. This sets the stage for Howe duality. In order to connect
with the Askey–Wilson algebra we take n = 3. The expressions of the operators K A
and K B acting on H that realize the relations (8) (together with the specific central
elements) are rather involved and we shall refer the reader to [10] for the formulas.
We shall only stress that these operators can be obtained in a dual way: They are
affinely related to the generators of the commutant of o q 1/2 (2) ⊕3 in o q 1/2 (6) as well
as to the intermediate U q (su(1, 1)) Casimir elements C (1234) = Δ (3) (C)⊗1⊗1 and
C (3456) = 1 ⊗ 1 ⊗ Δ (3) (C) of the q-metaplectic representation (see (9)). This can be
extended to higher ranks by letting n be arbitrary [11]. For n = 2 we are looking at
the Clebsch–Gordan problem for U q (su(1, 1)). The q-Hahn algebra that arises has
two dual realizations [12]: one as the commutant of o q 1/2 (2) ⊕2 in U q (u(4)) and the
other in terms of the following two U q (su(1, 1)) operators, (Δ(J 0 )⊗1⊗1)−(1⊗1⊗
Δ(J 0 )) and Δ (2) (C) (the full Casimir element) in the q-metaplectic representation.