Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials
Satoru Odake, Ryu Sasaki
备注：
7 pages; one reference added, published in Phys. Lett. B682 (2009) 130-136
Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical Hamiltonians, which are deformations of those for the Wilson and Askey-Wilson polynomials in terms of a degree \ell (\ell=1,2,...) eigenpolynomial. These polynomials are exceptional in the sense that they start from degree \ell\ge1 and thus not constrained by any generalisation of Bochner's theorem.
The Heun-Askey-Wilson algebra is introduced through generators $\{\boX,\boW\}$ and relations. These relations can be understood as an extension of the usual Askey-Wilson ones. A central element is given, and a canonical form of the Heun-Askey-Wilson algebra is presented. A homomorphism from the Heun-Askey-Wilson algebra to the Askey-Wilson one is identified. On the vector space of the polynomials in the variable $x=z+z^{-1}$, the Heun operator of Askey-Wilson type realizing $\boW$ can be characterized as the most general second order $q$-difference operator in the variable $z$ that maps polyno…
A construction is given of the most general representations of the q-oscillator algebra where both generators are tridiagonal. It is shown to be connected to the Askey-Wilson polynomials.
The $q$-difference equation, the shift and the contiguity relations of the Askey-Wilson polynomials are cast in the framework of the three and four-dimensional degenerate Sklyanin algebras $\mathfrak{ska}_3$ and $\mathfrak{ska}_4$. It is shown that the $q$-para Racah polynomials corresponding to a non-conventional truncation of the Askey-Wilson polynomials form a basis for a finite-dimensional representation of $\mathfrak{ska}_4$. The first order Heun operators defined by a degree raising condition on polynomials are shown to form a five-dimensional vector space that encompasses $\mathfrak{ska…
Automorphisms of the infinite dimensional Onsager algebra are introduced. Certain quotients of the Onsager algebra are formulated using a polynomial in these automorphisms. In the simplest case, the quotient coincides with the classical analog of the Askey-Wilson algebra. In the general case, generalizations of the classical Askey-Wilson algebra are obtained. The corresponding class of solutions of the non-standard classical Yang-Baxter algebra are constructed, from which a generating function of elements in the commutative subalgebra is derived. We provide also another presentation of the Ons…
The $sl_N$-Onsager algebra has been introduced by Uglov and Ivanov in 1995. In this letter, a FRT presentation of the $sl_N$-Onsager algebra is given, its current algebra and commutative subalgebra are constructed. Certain quotients of the $sl_N$-Onsager algebra are then considered, which produce `classical' analogs of higher rank extensions of the Askey-Wilson algebra. As examples, the cases $N=3$ and $N=4$ are described in details.
The higher rank Askey-Wilson algebra was recently constructed in the $n$-fold tensor product of $U_q(\mathfrak{sl}_2)$. In this paper we prove a class of identities inside this algebra, which generalize the defining relations of the rank one Askey-Wilson algebra. We extend the known construction algorithm by several equivalent methods, using a novel coaction. These allow to simplify calculations significantly. At the same time, this provides a proof of the corresponding relations for the higher rank $q$-Bannai-Ito algebra.
The Askey-Wilson algebra and its relatives such as the Racah and Bannai-Ito algebras were initially introduced in connection with the eponym orthogonal polynomials. They have since proved ubiquitous. In particular they admit presentations as commutants that are related through Howe duality. This paper surveys these results.
The Askey--Wilson algebras were used to interpret the algebraic structure hidden in the Racah--Wigner coefficients of the quantum algebra $U_q(\mathfrak{sl}_2)$. In this paper, we display an injection of a universal analog $\triangle_q$ of Askey--Wilson algebras into $U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)$ behind the application. Moreover we formulate the decomposition rules for $3$-fold tensor products of irreducible Verma $U_q(\mathfrak{sl}_2)$-modules and of finite-dimensional irreducible $U_q(\mathfrak{sl}_2)$-modules into the direct sums of finite-di…
The Askey-Wilson algebra is realized in terms of the elements of the quantum algebras $U_q(\mathfrak{su}(2))$ or $U_q(\mathfrak{su}(1,1))$. A new realization of the Racah algebra in terms of the Lie algebras $\mathfrak{su}(2)$ or $\mathfrak{su}(1,1)$ is given also. Details for different specializations are provided. The advantage of these new realizations is that one generator of the Askey-Wilson (or Racah) algebra becomes diagonal in the usual representation of the quantum algebras whereas the second one is tridiagonal. This allows to recover easily the recurrence relations of the associated …
The centralizer of the image of the diagonal embedding of $U_q(\mathfrak{sl}_2)$ in the tensor product of three irreducible representations is examined in a Schur-Weyl duality spirit. The aim is to offer a description in terms of generators and relations. A conjecture in this respect is offered with the centralizers presented as quotients of the Askey-Wilson algebra. Support for the conjecture is provided by an examination of the representations of the quotients. The conjecture is also shown to be true in a number of cases thereby exhibiting in particular the Temperley-Lieb, Birman-Murakami-We…
The occurrence of the Askey-Wilson (AW) algebra in the $SU(2)$ Chern-Simons (CS) theory and in the Reshetikhin-Turaev (RT) link invariant construction with quantum algebra $U_q(\mathfrak{su}_2)$ is explored. Tangle diagrams with three strands with some of them enclosed in a spin-$1/2$ closed loop are associated to the generators of the AW algebra. It is shown in both the CS theory and RT construction that the link invariant of these tangles obey the relations of the AW generators. It follows that the expectation values of certain Wilson loops in the CS theory satisfy relations dictated by the …
An operator of Heun-Askey-Wilson type is diagonalized within the framework of the algebraic Bethe ansatz using the theory of Leonard pairs. For different specializations and the generic case, the corresponding eigenstates are constructed in the form of Bethe states, whose Bethe roots satisfy Bethe ansatz equations of homogeneous or inhomogenous type. For each set of Bethe equations, an alternative presentation is given in terms of `symmetrized' Bethe roots. Also, two families of on-shell Bethe states are shown to generate two explicit bases on which a Leonard pair acts in a tridiagonal fashion…
A description of the embedding of the universal Askey--Wilson algebra, AW(3), in $U_q(sl_2)^{\otimes 3}$ is given in terms of the universal R-matrix of $U_q(sl_2)$. The generators of the centralizer of $U_q(sl_2)$ in its three-fold product are naturally expressed through conjugations of Casimir elements with R. They are identified as the images of the generators of AW(3) under the embedding map by showing that they obey the AW(3) relations. This is achieved by introducing a natural coaction also constructed with the help of the R-matrix.
A $q$-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the $2$-sphere, is obtained as the commutant of the $\mathfrak{o}_{q^{1/2}}(2) \oplus \mathfrak{o}_{q^{1/2}}(2)$ subalgebra of $\mathfrak{o}_{q^{1/2}}(4)$ in the $q$-oscillator representation of the quantized universal enveloping algebra $U_q(\mathfrak{u}(4))$. This $q$-Higgs algebra is also found as a specialization of the Askey--Wilson algebra embedded in the tensor product $U_q(\mathfrak{su}(1,1))\otimes U_q(\mathfrak{su}(1,1))$. The connection between these two approaches is establish…
The $q$-deformed Bannai-Ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra $\mathfrak{osp}_q(1\vert 2)$. It turned out to be isomorphic to the Askey-Wilson algebra. In the present paper these results will be extended to higher rank. The rank $n-2$ $q$-Bannai-Ito algebra $\mathcal{A}_n^q$, which by the established isomorphism also yields a higher rank version of the Askey-Wilson algebra, is constructed in the $n$-fold tensor product of $\mathfrak{osp}_q(1\vert 2)$. An explicit realization in terms of $q$-shift operators and reflections is proposed, …
A novel generalization of the Askey-Wilson algebra is presented and shown to be associated with coproducts in the quantum algebra $U_q(su(1,1))$. This algebra has 15 non-commuting generators given by $Q^{(A)}$, with $A\subset \{1,2,3,4\}$ and their 5 linearly independent inversions generated by the algebra automorphism $q\rightarrow q^{-1}$, $E\leftrightarrow F.$ The set of generators can be split into operators fixed under inversion, and those with an orientation under this inversion. We then show that the generators will either commute or satisfy q-commutator relations linear in the generato…
The universal Askey-Wilson algebra $AW(3)$ can be obtained as the commutant of $U_q(\mathfrak{su}(1,1))$ in $U_q(\mathfrak{su}(1,1))^{\otimes3}$. We analyze the commutant of $\mathfrak{o}_{q^{1/2}}(2)\oplus\mathfrak{o}_{q^{1/2}}(2)\oplus\mathfrak{o}_{q^{1/2}}(2)$ in $q$-oscillator representations of $\mathfrak{o}_{q^{1/2}}(6)$ and show that it also realizes $AW(3)$. These two pictures of $AW(3)$ are shown to be dual in the sense of Howe; this is made clear by highlighting the role of the intermediate Casimir elements of each members of the dual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{…
CNRS, IRIF Université Paris Cité（法国国家科学研究中心，IRIF巴黎西岱大学）；Department of Mathematics, University of California Berkeley（加州大学伯克利分校数学系）；Mathematical Institute, University of Oxford（牛津大学数学研究所）；Nu Quantum Ltd.（Nu量子有限公司）；Department of Mathematics, Sungkyunkwan University (SKKU)（成均馆大学（SKKU）数学系）
本文为 $q$-Askey 方案和 Askey 方案中的每个正交多项式族建立了以 lecture hall 图上的路径表示的混合矩和系数的组合模型，推广了 Corteel 和 Kim 关于小 $q$-Jacobi 多项式的结果，并首次给出了 Askey-Wilson 多项式参数对称性的组合证明。
In previous papers, we discussed the recurrence relations of the multi-indexed orthogonal polynomials of the Laguerre, Jacobi, Wilson and Askey-Wilson types. In this paper we explore those of the Racah and $q$-Racah types. For the $M$-indexed ($q$-)Racah polynomials, we derive $3+2M$ term recurrence relations with variable dependent coefficients and $1+2L$ term ($L\geq M+1$) recurrence relations with constant coefficients. Based on the latter, the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi-indexed ($q$-)Rac…
We show that intertwining operators for the discrete Fourier transform form a cubic algebra $\mathcal{C}_q$ with $q$ a root of unity. This algebra is intimately related to the two other well-known realizations of the cubic algebra: the Askey-Wilson algebra and the Askey-Wilson-Heun algebra.
Via the solutions of systems of algebraic equations of Bethe Ansatz type, we arrive at bounds for the zeros of orthogonal (basic) hypergeometric polynomials belonging to the Askey-Wilson, Wilson and continuous Hahn families.
The theory of contractions of 2D 2nd order quantum superintegrable systems and its relation to the Askey scheme for hypergeometric orthogonal polynomials
Upon solving a finite discrete reduction of the difference Heun equation, we arrive at an elliptic generalization of the Racah polynomials. We exhibit the three-term recurrence relation and the orthogonality relations for these elliptic Racah polynomials. The well-known $q$-Racah polynomials of Askey and Wilson are recovered as a trigonometric limit.
