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

Abstract

The higher rank Racah algebra $R(n)$ introduced recently 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 $Z_n({\mathfrak{sl}}_2)$ of the diagonal embedding of $U({\mathfrak{sl}}_2)$ in $U({\mathfrak{sl}}_2{)}^{\otimes n}$. This leads to a first and novel presentation of the centralizer $Z_n({\mathfrak{sl}}_2)$ 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.

References in corpus (7)

• The Askey-Wilson algebra and its avatars
• Embedding of the Racah Algebra R($\mathbf{n}$) and Superintegrability
• Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of $U_q({\mathfrak{sl}}_2)$
• The general Racah algebra as the symmetry algebra of generic systems on pseudo--spheres
• $N$-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra $\mathcal{S}\mathcal{W}(N)$
• New realizations of algebras of the Askey-Wilson type in terms of Lie and quantum algebras
• The rational Sklyanin algebra and the Wilson and para-Racah polynomials

Cited by in corpus (8)

• Representations of the rank two Racah algebra and orthogonal multivariate polynomials
• Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras
• The missing label of ${\mathfrak{su}}_3$ and its symmetry
• Polynomial algebras from commutants: Classical and Quantum aspects of ${\mathcal{A}}_3$
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• Generalized quadratic commutator algebras of PBW-type
• $\lambda$-Griffiths polynomials: Bispectrality and biorthogonality
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions