The Racah Algebra and sl n
211
A full account of R n is presented in [4]. We mention one fact here: the operators
C A are not linearly independent. By formula (17) in [4] we have
C A =
{i,j }⊂A
C ij − (|A| − 2)
i∈A
C i .
Hence, if one wants to present realizations of R n , it suffices to give expressions for
the operators C ij and C i . In Sect. 4 we will present the higher rank Racah algebra
as given in [5] this way.
3 Realizing sl n in n − 1 Variables
Let sl n (R) be the algebra of n × n matrices whose trace equals zero and with the
commutator as Lie bracket. Let E ij be the matrix whose entries are equal to 0 except
for the entry on the ith row and j th column which equals 1. Then the Lie algebra
sl n (R) is generated by the set
{E ij |1 ≤ i, j ≤ n and i = j } ∪ {E ii − E nn |1 ≤ i ≤ n − 1}.
Let u i , i ∈ {1 . . . n − 1} be real variables. We introduce the differential operators:
T ij := −u j ∂ i
i = j and i, j < n
T in := −∂ i
i < n
T nj := u j ˜
E
j < n
˜
T d := −u d ∂ d − ˜
E
d < n,
where the operator ˜
E is defined as
˜
E :=
n−1
i=1
u i ∂ i − k.
Using again the commutator as Lie bracket we denote by D n the Lie algebra spanned
by all the ˜
T d and T ij . The real number k is a deformation parameter that leaves the
algebra relations invariant. One observes that sl n (R) and D n are isomorphic. The
isomorphism σ is given by
σ (E ij ) = T ij
σ (E dd − E nn ) = ˜
T d .
Note that this isomorphism does not extend to their universal enveloping algebras.