212
H. De Bie et al.
3.1 Some Operators in U (D n )
We introduce a number of operators we need later on. Observe that the operator ˜
E
is in the universal enveloping algebra U (D n ) but not in D n because D n lacks the
identity:
˜
E =
−1
n
k +
n−1
d=1
˜
T d
.
Let u B :=
k∈B u k . We have
u B ˜
E :=
j ∈B
T nj .
We will also express u B ∂ α in function of the generators:
u B ∂ α := −δ αB ( ˜
T α + ˜
E) −
j ∈B\α
T αj ,
(1)
where we introduced a new symbol standing for:
δ αB :=
0, if α /
∈ B
1, if α ∈ B.
It is then easy to check the following Lemma.
Lemma 1 The following holds
[u B ˜
E, ∂ α ] = −u B ∂ α − δ αB ˜
E
[u A , ∂ α ] = −δ αA .
4 Realization of R n in n − 2 Variables
In [5] an explicit differential operator realization of R n was given in Theorem 5. We
repeat this theorem here.
Theorem 1 The space Π
n−2
k
of all polynomials of degree k in n − 2 variables
carries a realization of the rank n − 2 Racah algebra R n . This realization is given
explicitly by
C i = ν i (ν i − 1),
i ∈ [n]