230
J. Gaboriaud et al.
be checked directly, again abusing notation, that M 1 =
1
2 (N 1 + N 2 − N 3 − N 4 )
and M 2 = −
1
4
L 12
2 + L 34
2 + L 13
2 + L 23
2 + L 14
2 + L 24
2
satisfy the relations
given in Eq. (4) with δ 1 = −
1
2 (N 1 + N 2 + N 3 + N 4 + 2)(L 12
2 − L 34
2 ) and
δ 2 =
1
2 (N 1 + N 2 + N 3 + N 4 + 2) 2 − (L 12
2 + L 34
2 + 2), in correspondence with
the preceding expressions for δ 1 and δ 2 in the realization J (1234)
•
of su(1, 1). From
the expressions of these last M 1 and M 2 , we can claim that the Hahn algebra is the
commutant of o(2) ⊕ o(2) in U (u(4)) represented on H. Let us stress that it is the
universal enveloping algebra of u(4) that intervenes here.
4.2 The Bannai–Ito Ensemble
The Bannai–Ito algebra [7] takes its name after the Bannai–Ito polynomials that
enter in the Racah coefficients of the Lie superalgebra osp(1|2). This algebra has
three generators K i , i = 1, . . . , 3 that satisfy the relations
{K i , K j } = K k + ω k ,
i = j = k ∈ {1, 2, 3}
(5)
with ω i central and {X, Y } = XY + Y X. The relevant reductive pair in this
case is (o(6), osp(1|2)) and the representation space H is that of Dirac spinors in
six dimensions with the Clifford algebra generated by the elements γ μ verifying
{γ μ , γ ν } = −2δ μν , μ, ν = 1, . . . , 6. That the pair (o(6), osp(1|2)) is dually
represented on H is seen as follows: The spinorial representation of o(6) with
generators
J μν = −iL μν + Σ μν ,
L μν = x μ ∂ ν − x ν ∂ μ ,
Σ μν =
i
2
γ μ γ ν
(6)
leaves invariant the following operators:
J − = −i
1≤μ≤6
γ μ ∂ μ ,
J + = −i
1≤μ≤6
γ μ x μ ,
J 0 =
1≤μ≤6
x μ ∂ μ ,
(7)
which in turn realize the commutation relations of the Lie superalgebra osp(1|2):
[J 0 , J ± ] = ±J ± , {J + , J − } = −2J 0 with J 0 even and J ± odd. Howe duality
thus takes place. As a matter of fact, for any subset A ⊂ {1, . . . , 6} of cardinality |A| the operators J A
− = −i
μ∈A γ μ ∂ μ , J A
+ = −i
μ∈A γ μ x μ , and
J A
0 =
|A|
2 +
μ∈A x μ ∂ μ realize osp(1|2). The Casimir element of osp(1|2) is given
by C =
1
2 ([J − , J + ] − 1)S with S the grade involution obeying S 2 = 1, [S, J 0 ] = 0,
{S, J ± } = 0. In the realizations at hand, S A = i |A|/2
μ∈A γ μ with |A| even.
J. Gaboriaud et al.
be checked directly, again abusing notation, that M 1 =
1
2 (N 1 + N 2 − N 3 − N 4 )
and M 2 = −
1
4
L 12
2 + L 34
2 + L 13
2 + L 23
2 + L 14
2 + L 24
2
satisfy the relations
given in Eq. (4) with δ 1 = −
1
2 (N 1 + N 2 + N 3 + N 4 + 2)(L 12
2 − L 34
2 ) and
δ 2 =
1
2 (N 1 + N 2 + N 3 + N 4 + 2) 2 − (L 12
2 + L 34
2 + 2), in correspondence with
the preceding expressions for δ 1 and δ 2 in the realization J (1234)
•
of su(1, 1). From
the expressions of these last M 1 and M 2 , we can claim that the Hahn algebra is the
commutant of o(2) ⊕ o(2) in U (u(4)) represented on H. Let us stress that it is the
universal enveloping algebra of u(4) that intervenes here.
4.2 The Bannai–Ito Ensemble
The Bannai–Ito algebra [7] takes its name after the Bannai–Ito polynomials that
enter in the Racah coefficients of the Lie superalgebra osp(1|2). This algebra has
three generators K i , i = 1, . . . , 3 that satisfy the relations
{K i , K j } = K k + ω k ,
i = j = k ∈ {1, 2, 3}
(5)
with ω i central and {X, Y } = XY + Y X. The relevant reductive pair in this
case is (o(6), osp(1|2)) and the representation space H is that of Dirac spinors in
six dimensions with the Clifford algebra generated by the elements γ μ verifying
{γ μ , γ ν } = −2δ μν , μ, ν = 1, . . . , 6. That the pair (o(6), osp(1|2)) is dually
represented on H is seen as follows: The spinorial representation of o(6) with
generators
J μν = −iL μν + Σ μν ,
L μν = x μ ∂ ν − x ν ∂ μ ,
Σ μν =
i
2
γ μ γ ν
(6)
leaves invariant the following operators:
J − = −i
1≤μ≤6
γ μ ∂ μ ,
J + = −i
1≤μ≤6
γ μ x μ ,
J 0 =
1≤μ≤6
x μ ∂ μ ,
(7)
which in turn realize the commutation relations of the Lie superalgebra osp(1|2):
[J 0 , J ± ] = ±J ± , {J + , J − } = −2J 0 with J 0 even and J ± odd. Howe duality
thus takes place. As a matter of fact, for any subset A ⊂ {1, . . . , 6} of cardinality |A| the operators J A
− = −i
μ∈A γ μ ∂ μ , J A
+ = −i
μ∈A γ μ x μ , and
J A
0 =
|A|
2 +
μ∈A x μ ∂ μ realize osp(1|2). The Casimir element of osp(1|2) is given
by C =
1
2 ([J − , J + ] − 1)S with S the grade involution obeying S 2 = 1, [S, J 0 ] = 0,
{S, J ± } = 0. In the realizations at hand, S A = i |A|/2
μ∈A γ μ with |A| even.