The Racah Algebra and sl n
213
and, for i, j ∈ {3, . . . , n},
C 12 = −
k − 1 −
n−2
=1
u ∂ u
−k − ∂ u 1 +
n−2
=1
u ∂ u
+ 2ν 2
k −
n−2
=1
u ∂ u
− 2ν 1
−k − ∂ u 1 +
n−2
=1
u ∂ u
+ (ν 1 + ν 2 )(ν 1 + ν 2 − 1)
C 1j = −
⎛
⎝ 1 −
j −2
=1
u
⎞
⎠
2
k − 1 −
n−2
=1
u ∂ u
∂ u j −2 − ∂ u j −1
+ 2ν j
⎛
⎝ 1 −
j −2
=1
u
⎞
⎠
k −
n−2
=1
u ∂ u
− 2ν 1
⎛
⎝ 1 −
j −2
=1
u
⎞
⎠
∂ u j −2 − ∂ u j −1
+ (ν 1 + ν j )(ν 1 + ν j − 1)
C 2j = −
⎛
⎝
j −2
=1
u
⎞
⎠
2
1 − k − ∂ u 1 +
n−2
=1
u ∂ u
∂ u j −2 − ∂ u j −1
+ 2ν j
⎛
⎝
j −2
=1
u
⎞
⎠
k + ∂ u 1 −
n−2
=1
u ∂ u
+ 2ν 2
⎛
⎝
j −2
=1
u
⎞
⎠
∂ u j −2 − ∂ u j −1
+ (ν 2 + ν j )(ν 2 + ν j − 1)
C ij = −
⎛
⎝
i−2
=j −1
u
⎞
⎠
2
∂ u i−2 − ∂ u i−1
∂ u j −2 − ∂ u j −1
+ 2ν j
⎛
⎝
i−2
=j −1
u
⎞
⎠
∂ u i−2 − ∂ u i−1
− 2ν i
⎛
⎝
i−2
=j −1
u
⎞
⎠
∂ u j −2 − ∂ u j −1
+ (ν i + ν j )(ν i + ν j − 1),
where we assume i > j and with u n−1 = 0 whenever it appears.
We want to express these operators as elements in U (D n−1 ).