Racah algebras, the centralizer $Z_n(\mathfrak{sl}_2)$ and its Hilbert-Poincaré series

Abstract

The higher rank Racah algebra R(n) introduced in [1] is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra sR(n), is then introduced. Using results from classical invariant theory, this sR(n) algebra is shown to be isomorphic to the centralizer Z n (sl 2 ) of the diagonal embedding of U (sl 2 ) in U (sl 2 ) ⊗n . This leads to a first and novel presentation of the centralizer Z n (sl 2 ) in terms of generators and defining relations. An explicit formula of its Hilbert-Poincaré series is also obtained and studied. The extension of the results to the study of the special Askey-Wilson algebra and its higher rank generalizations is discussed.

The missing label of $\mathfrak{su}_3$ and its symmetry

“…Remark 1.2. In the representation M m of Z 2 (su 3 ), the Casimir elements k 1 , k 2 , k 3 , l 1 , l 2 , l 3 , act as numbers, see (12). The couple of numbers for (k 1 , l 1 ) (resp.…”

“…The last relation ( 52) is the analogue of the relation that one must add to the Racah algebra to obtain the centraliser of the diagonal su 2 in U (su 2 ) ⊗3 (see e.g. [9] or [12] for the higher Racah algebras). Note that the left hand side of ( 52) is central in the algebra defined by Relations (51).…”

Bethe ansatz diagonalization of the Heun-Racah operator

“…The Racah polynomials are at the top of the Askey scheme and provide through different limits all the discrete hypergeometric polynomials. Its recurrence and difference relations satisfy the Racah algebra introduced in the algebraic study of the 6j-symbol and of the Racah problem for su(2) [33,29] (see [20] for a review). The Racah algebra can be embedded in sl 2 [32,41,14,22], is associated to the DAHA (C ∨ 1 , C 1 ) [36] and appears in the context of superintegrable models [38,45,39,30,31].…”