The Racah Algebra and sl n
215
In line 3 we used Lemma 1. Let
L
(j )
3 := u [j −2]
∂ j −2 − ∂ j −1
L
(j )
4 := u [j −2]
−∂ 1 + ˜
E
.
Both L
(j )
3 and L
(j )
4 can be expressed in function of the generators of U (D n−1 ), again
because of expression (1). The operator
C 2j can be expressed as follows:
C 2j = −L
(j )
3 L
(j )
4 − (2ν j − 1)L
(j )
4 + 2ν 2 L
(j )
3 + (ν 2 + ν j )(ν 2 + ν j − 1).
This means that
C 2j is also in U (D n−1 ).
Consider the first term of the operator
C ij :
− u
2
[j −1,i−2] (∂ i−2 − ∂ i−1 )
∂ j −2 − ∂ j −1
= −u [j −1,i−2]
(∂ i−2 − ∂ i−1 ) u [j −1,i−2] − 1
∂ j −2 − ∂ j −1
= −u [j −1,i−2] (∂ i−2 − ∂ i−1 ) u [j −1,i−2]
∂ j −2 − ∂ j −1
+ u [j −1,i−2]
∂ j −2 − ∂ j −1
.
In line 2 we used Lemma 1. Let
L
(ij )
5 := u [j −1,i−2] (∂ i−2 − ∂ i−1 )
L
(ij )
6 := u [j −1,i−2]
∂ j −2 − ∂ j −1
.
Both L
(ij )
5 and L
(ij )
6 can be expressed in function of the generators of U (D n−1 ). The
operator
C ij can be expressed as follows:
C ij = −L
(ij )
5 L
(ij )
6 − (2ν i − 1)L
(ij )
6 + 2ν j L
(ij )
5 + (ν i + ν j )(ν i + ν j − 1).
This means that
C ij is also in U (D n−1 ).This proves Conjecture 1 for this differential
realization.
5 Conclusions
In this paper we have considered the higher rank Racah algebra in one of its
differential operator realizations, obtained in the recent paper [5]. We have shown
that this realization can be embedded in the enveloping algebra of a differential
operator realization of sl n verifying Conjecture 1 in this realization.