Related papers: Polynomial algebras from commutants: Classical and…
Construction of superintegrable systems based on Lie algebras have been introduced over the years. However, these approaches depend on explicit realisations, for instance as a differential operators, of the underlying Lie algebra. This is…
Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial…
The rank-$1$ Racah algebra $R(3)$ plays a pivotal role in the theory of superintegrable systems. It appears as the symmetry algebra of the $3$-parameter system on the $2$-sphere from which all second-order conformally flat superintegrable…
Recent results on the Racah algebra $\mathcal{R}_n$ of rank $n - 2$ are reviewed. $\mathcal{R}_n$ is defined in terms of generators and relations and sits in the centralizer of the diagonal action of $\mathfrak{su}(1,1)$ in…
The recent interest in the study of higher-rank polynomial algebras related to $n$-dimensional classical and quantum superintegrable systems with coalgebra symmetry and their connection with the generalised Racah algebra $R(n)$, a…
The universal character of the Racah algebra will be illustrated by showing that it is at the center of the relations between the Racah polynomials, the recoupling of three su(1,1) representations and the symmetries of the generic…
The Racah algebra encodes the bispectrality of the eponym polynomials. It is known to be the symmetry algebra of the generic superintegrable model on the $2$-sphere. It is further identified as the commutant of the $\mathfrak{o}(2) \oplus…
The connection between the recoupling scheme of four copies of $\mathfrak{su}(1,1)$, the generic superintegrable system on the 3 sphere, and bivariate Racah polynomials is identified. The Racah polynomials are presented as connection…
In this work, we refine recent results on the explicit construction of polynomial algebras associated with commutants of subalgebras in enveloping algebras of Lie algebras by considering an additional grading with respect to the subalgebra.…
The Racah algebra $R(n)$ of rank $(n-2)$ is obtained as the commutant of the \mbox{$\mathfrak{o}(2)^{\oplus n}$} subalgebra of $\mathfrak{o}(2n)$ in oscillator representations of the universal algebra of $\mathfrak{o}(2n)$. This result is…
We characterize the symmetry algebra of the generic superintegrable system on a pseudo-sphere corresponding to the homogeneous space $SO(p,q+1)/SO(p,q)$ where $p+q={\cal N}$, ${\cal N}\in\mathbb N$. We show that this algebra is independent…
The analysis of the most general second-order superintegrable system in two dimensions: the generic 3-parameter model on the 2-sphere, is cast in the framework of the Racah problem for the su(1,1) algebra. The Hamiltonian of the 3-parameter…
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…
We consider the generic quantum superintegrable system on the $d$-sphere with potential $V(y)=\sum_{k=1}^{d+1}\frac{b_k}{y_k^2}$, where $b_k$ are parameters. Appropriately normalized, the symmetry operators for the Hamiltonian define a…
The Racah algebra, a quadratic algebra with two independent generators, is central in the analysis of superintegrable models and encodes the properties of the Racah polynomials. It is the algebraic structure behind the su(1,1) Racah problem…
It has been known since 2007 that the Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere,…
Embeddings of the Racah algebra into the Bannai-Ito algebra are proposed in two realizations. First, quadratic combinations of the Bannai-Ito algebra generators in their standard realization on the space of polynomials are seen to generate…
One of spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group $SU_q(2)$ through the Askey-Wilson polynomials, associated with the $q$-hypergeometric functions…
A higher rank generalization of the (rank one) Racah algebra is obtained as the symmetry algebra of the Laplace-Dunkl operator associated to the $\mathbb{Z}_2^n$ root system. This algebra is also the invariance algebra of the generic…
The goal of the present paper is to provide a detailed study of irreducible representations of the algebra generated by the symmetries of the generic quantum superintegrable system on the $d$-sphere. Appropriately normalized, the symmetry…