Askey-Wilson Algebra Research Articles

The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms

We propose a definition by generators and relations of the rank $n-2$ Askey-Wilson algebra $\mathfrak{aw}(n)$ for any integer $n$, generalising the known presentation for the usual case $n=3$. The generators are indexed by connected subsets of $\{1,\dots,n\}$ and the simple and rather small set of defining relations is directly inspired from the known case of $n=3$. Our first main result is to prove the existence of automorphisms of $\mathfrak{aw}(n)$ satisfying the relations of the braid group on $n+1$ strands. We also show the existence of coproduct maps relating the algebras for different values of $n$. An immediate consequence of our approach is that the Askey-Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the $n$-fold tensor product of the quantum group ${\rm U}_q(\mathfrak{sl}_2)$ or, equivalently, onto the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. We also obtain a family of central elements of the Askey-Wilson algebras which are shown, as a direct by-product of our construction, to be sent to $0$ in the realisation in the $n$-fold tensor product of ${\rm U}_q(\mathfrak{sl}_2)$, thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.

Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{\omega N}$ for some nonzero real parameter $\omega$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.

The Clebsch-Gordan Rule for U(sl<sub>2</sub>), the Krawtchouk Algebras and the Hamming Graphs

Let $D\geq 1$ and $q\geq 3$ be two integers. Let $H(D)=H(D,q)$ denote the $D$-dimensional Hamming graph over a $q$-element set. Let ${\mathcal T}(D)$ denote the Terwilliger algebra of $H(D)$. Let $V(D)$ denote the standard ${\mathcal T}(D)$-module. Let $\omega$ denote a complex scalar. We consider a unital associative algebra $\mathfrak K_\omega$ defined by generators and relations. The generators are $A$ and $B$. The relations are $A^2 B-2 ABA +B A^2 =B+\omega A$, $B^2A-2 BAB+AB^2=A+\omega B$. The algebra $\mathfrak K_\omega$ is the case of the Askey-Wilson algebras corresponding to the Krawtchouk polynomials. The algebra $\mathfrak K_\omega$ is isomorphic to ${\rm U}(\mathfrak{sl}_2)$ when $\omega^2\not=1$. We view $V(D)$ as a $\mathfrak{K}_{1-\frac{2}{q}}$-module. We apply the Clebsch-Gordan rule for ${\rm U}(\mathfrak{sl}_2)$ to decompose $V(D)$ into a direct sum of irreducible ${\mathcal T}(D)$-modules.

An Askey-Wilson Algebra of Rank 2

An algebra is introduced which can be considered as a rank 2 extension of the Askey-Wilson algebra. Relations in this algebra are motivated by relations between coproducts of twisted primitive elements in the two-fold tensor product of the quantum algebra $\mathcal{U}_{q}(\mathfrak{sl}(2,\mathbb C))$. It is shown that bivariate $q$-Racah polynomials appear as overlap coefficients of eigenvectors of generators of the algebra. Furthermore, the corresponding $q$-difference operators are calculated using the defining relations of the algebra, showing that it encodes the bispectral properties of the bivariate $q$-Racah polynomials.

Chern–Simons theory, link invariants and the Askey–Wilson algebra

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 Uq(su2) 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 AW algebra and that the link invariants do not distinguish the corresponding linear combinations of links.

Racah Algebras, the Centralizer $$Z_n({{{\mathfrak {s}}}{{\mathfrak {l}}}}_2)$$ and Its Hilbert–Poincaré Series

The higher rank Racah algebra R(n) introduced in De Bie et al. (J Phys A Math Theor 51(2):025203, 2017. arXiv:1610.02638 ) is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra sR(n), is then introduced. Using the results from classical invariant theory, this sR(n) algebra is shown to be isomorphic to the centralizer $$Z_{n}({{\mathfrak {s}}}{{\mathfrak {l}}}_2)$$ of the diagonal embedding of $$U({{\mathfrak {s}}}{{\mathfrak {l}}}_2)$$ in $$U({{\mathfrak {s}}}{{\mathfrak {l}}}_2)^{\otimes n}$$ . This leads to a first and novel presentation of the centralizer $$Z_{n}({{\mathfrak {s}}}{{\mathfrak {l}}}_2)$$ 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.

An algebraic interpretation of the intertwining operators associated with the discrete Fourier transform

We show that intertwining operators for the discrete Fourier transform form a cubic algebra Cq, with q being a root of unity. This algebra is intimately related to the other two well-known realizations of the cubic algebra: the Askey–Wilson algebra and the Askey–Wilson–Heun algebra.

Temperley–Lieb, Birman–Murakami–Wenzl and Askey–Wilson Algebras and Other Centralizers of $$U_q(\mathfrak {sl}_2)$$

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–Wenzl and one-boundary Temperley–Lieb algebras as quotients of the Askey–Wilson algebra.

A conjecture concerning the q-Onsager algebra

The q-Onsager algebra Oq is defined by two generators W0,W1 and two relations called the q-Dolan/Grady relations. Recently Baseilhac and Kolb obtained a PBW basis for Oq with elements denoted{Bnδ+α0}n=0∞,{Bnδ+α1}n=0∞,{Bnδ}n=1∞. In their recent study of a current algebra Aq, Baseilhac and Belliard conjecture that there exist elements{W−k}k=0∞,{Wk+1}k=0∞,{Gk+1}k=0∞,{G˜k+1}k=0∞ in Oq that satisfy the defining relations for Aq. In order to establish this conjecture, it is desirable to know how the elements on the second displayed line above are related to the elements on the first displayed line above. In the present paper, we conjecture the precise relationship and give some supporting evidence. This evidence consists of some computer checks on SageMath due to Travis Scrimshaw, a proof of the analog conjecture for the Onsager algebra O, and a proof of our conjecture for a homomorphic image of Oq called the universal Askey-Wilson algebra.

Finite-dimensional irreducible modules of the universal Askey–Wilson algebra at roots of unity

Let F denote an algebraically closed field and assume that q∈F is a primitive dth root of unity with d≠1,2,4. The universal Askey–Wilson algebra △q is a unital associative F-algebra defined by generators and relations. The generators are A,B,C and the relations assert that each ofA+qBC−q−1CBq2−q−2,B+qCA−q−1ACq2−q−2,C+qAB−q−1BAq2−q−2 commutes with A,B,C. We show that every finite-dimensional irreducible △q-module is of dimension less than or equal to{difdis odd;d/2ifdis even. Moreover we provide an example to show that the bound is tight.

The universal DAHA of type and Leonard triples

Assume that is an algebraically closed field and q is a nonzero scalar in that is not a root of unity. The universal Askey–Wilson algebra is a unital associative -algebra generated by A, B, C and the relations state that each of is central in The universal DAHA of type is a unital associative -algebra generated by and the relations state that It was given an -algebra homomorphism that sends Therefore, any -module can be considered as a -module. Let V denote a finite-dimensional irreducible -module. In this paper, we show that A, B, C are diagonalizable on V if and only if A, B, C act as Leonard triples on all composition factors of the -module V.

The Heun–Racah and Heun–Bannai–Ito algebras

The Heun–Racah and Heun–Bannai–Ito algebras are introduced. Specializations of these algebras are seen to be realized by the operators obtained by applying the algebraic Heun construct to the bispectral operators of the Racah and Bannai–Ito polynomials. The study supplements the results on the Heun–Askey–Wilson algebra and completes the description of the Heun algebras associated with the polynomial families at the top of the Askey scheme, its q-analog, and the Bannai–Ito one.

New realizations of algebras of the Askey–Wilson type in terms of Lie and quantum algebras

The Askey–Wilson algebra is realized in terms of the elements of the quantum algebras [Formula: see text] or [Formula: see text]. A new realization of the Racah algebra in terms of the Lie algebras [Formula: see text] or [Formula: see text] is also given. 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 us to recover easily the recurrence relations of the associated orthogonal polynomials of the Askey scheme. These realizations involve rational functions of the Cartan generator of the quantum algebras, where they are linear with respect to the other generators and depend on the Casimir element of the quantum algebras.

The dual pair (Uq(su(1,1)),oq1/2(2n)), q-oscillators, and Askey-Wilson algebras

The universal Askey–Wilson algebra AW(3) can be obtained as the commutant of Uq(su(1,1)) in Uq(su(1,1))⊗3. We analyze the commutant of oq1/2(2)⊕oq1/2(2)⊕oq1/2(2) in q-oscillator representations of oq1/2(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 member of the dual pair Uq(su(1,1)),oq1/2(6). We also generalize these results. A higher rank extension of the Askey–Wilson algebra denoted AW(n) can be defined as the commutant of Uq(su(1,1)) in Uq(su(1,1))⊗n, and a dual description of AW(n) as the commutant of oq1/2(2)⊕n in q-oscillator representations of oq1/2(2n) is offered by calling upon the dual pair Uq(su(1,1)),oq1/2(2n).

Revisiting the Askey–Wilson algebra with the universal R-matrix of

A description of the embedding of a centrally extended Askey–Wilson algebra, , in is given in terms of the universal R-matrix of . The generators of the centralizer of in its three-fold tensor product are naturally expressed through conjugations of Casimir elements with R. They are seen as the images of the generators of under the embedding map by showing that they obey the relations. This is achieved by introducing a natural coaction also constructed with the help of the R-matrix.

Higher Rank Relations for the Askey-Wilson and q-Bannai-Ito Algebra

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.

Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz

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. In a second part, the (in)homogeneous Baxter T-Q relations are derived. Certain realizations of the Heun-Askey-Wilson operator as second q-difference operators are introduced. Acting on the Q-polynomials, they produce the T-Q relations. For a special case, the Q-polynomial is identified with the Askey-Wilson polynomial, which allows one to obtain the solution of the associated Bethe ansatz equations. The analysis presented can be viewed as a toy model for studying integrable models generated from the Askey-Wilson algebra and its generalizations. As examples, the q-analog of the quantum Euler top and various types of three-sites Heisenberg spin chains in a magnetic field with inhomogeneous couplings, three-body and boundary interactions are solved. Numerical examples are given. The results also apply to the time-band limiting problem in signal processing.

The Higher Rank q-Deformed Bannai-Ito and Askey-Wilson Algebra

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, which will be called the $\mathbb{Z}_2^n$ $q$-Dirac-Dunkl model. The algebra $\mathcal{A}_n^q$ is shown to arise as the symmetry algebra of the constructed $\mathbb{Z}_2^n$ $q$-Dirac-Dunkl operator and to act irreducibly on modules of its polynomial null-solutions. An explicit basis for these modules is obtained using a $q$-deformed $\mathbf{CK}$-extension and Fischer decomposition.

The Heun–Askey–Wilson Algebra and the Heun Operator of Askey–Wilson Type

The Heun–Askey–Wilson algebra is introduced through generators $$\{{\textsf {X}},{\textsf {W}}\}$$ 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 $${\textsf {W}}$$ can be characterized as the most general second-order q-difference operator in the variable z that maps polynomials of degree n in $$x=z+z^{-1}$$ into polynomials of degree $$n+1$$ .

The q-Higgs and Askey–Wilson algebras

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 oq1/2(2)⊕oq1/2(2) subalgebra of oq1/2(4) in the q-oscillator representation of the quantized universal enveloping algebra Uq(u(4)). This q-Higgs algebra is also found as a specialization of the Askey–Wilson algebra embedded in the tensor product Uq(su(1,1))⊗Uq(su(1,1)). The connection between these two approaches is established on the basis of the Howe duality of the pair (oq1/2(4),Uq(su(1,1))).