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The dual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(2n)\big)$, $q$-oscillators and Askey-Wilson algebras

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Pith's one-line read The Askey-Wilson algebra also appears as the commutant of a stack of q-deformed rotation algebras in q-oscillator space, dual to its usual tensor-product realization.

desk verdict A clean and likely new n=3 dual realization of AW(3) via q-oscillators and Howe duality, with a higher-rank AW(n) generalization that is plausible but rests on unproved commutant-generation claims. read the letter →

arxiv 1908.04277 v1 pith:5SYKXL7N submitted 2019-08-12 math-ph math.MP

Luc Frappat , Julien Gaboriaud , Eric Ragoucy , Luc Vinet This is my paper · ORCID
classification math-phmath.MP MSC 17B3781R5033D45
keywords Askey-WilsonalgebraHowedualityq-oscillatorrepresentationU_q(su(11))o_{q^{1/2}}(2n)commutanthigher-rankq-deformeddualpair
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the Askey-Wilson algebra, the algebraic structure behind Askey-Wilson polynomials, can be realized in two apparently different ways that are dual to each other. The standard picture describes it through the recoupling of three or more copies of the quantum algebra $U_q(\mathfrak{su}(1,1))$. The new picture obtains the same algebra as the set of operators commuting with a stack of $q$-deformed rotation algebras $\mathfrak{o}_{q^{1/2}}(2)^{\oplus n}$ inside a $q$-oscillator representation of $\mathfrak{o}_{q^{1/2}}(2n)$. The bridge is the $q$-deformed Howe dual pair between $U_q(\mathfrak{su}(1,1))$ and $\mathfrak{o}_{q^{1/2}}(2n)$, whose Casimir elements are shown to be affinely related. If correct, the result gives the higher-rank Askey-Wilson algebra $AW(n)$ a second, oscillator-based description and extends the classical Racah algebra duality to the $q$-deformed setting.

What carries the argument

The load-bearing object is the $q$-deformed Howe dual pair $(U_q(\mathfrak{su}(1,1)), \mathfrak{o}_{q^{1/2}}(2n))$ acting on the $2n$ $q$-oscillator Hilbert space, together with the affine Casimir correspondence of Eq. (5.9). A dual pair here is a pair of algebras whose actions commute and whose irreducible representations pair up; the paper uses the $q$-deformed version established in the references. The dual-pair structure guarantees that the two algebras commute, and the affine correspondence then identifies the intermediate Casimir elements of $U_q(\mathfrak{su}(1,1))$ with the quadratic Casimir-type elements $\Lambda_{[i;j]}$ of $\mathfrak{o}_{q^{1/2}}(2n)$. Since $AW(n)$ is generated by the $U_q(\mathfrak{su}(1,1))$ intermediate Casimirs, the same affine map exhibits the $\mathfrak{o}_{q^{1/2}}(2)^{\oplus n}$-commutant as a generating set for the same algebra.

What would settle it

Compute the full commutant of $\{L_{12},L_{34},L_{56}\}$ in the $q$-oscillator representation of $\mathfrak{o}_{q^{1/2}}(6)$; if it contains any element outside the algebra generated by the six $\Lambda$'s of Eq. (4.1), the identification with $AW(3)$ fails. The same direct computation for $n=4$ with the ten $\Lambda$'s would test the higher-rank claim.

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Extended reading notes

Core claim

The paper's central claim is that for every $n$, the commutant of $\mathfrak{o}_{q^{1/2}}(2)^{\oplus n}$ in the $q$-oscillator representation of $\mathfrak{o}_{q^{1/2}}(2n)$ is generated by the elements $\Lambda_i = (L_{2i-1,2i})^2$ and the $\Lambda_{ij}$ built from quadratic Casimir pieces, and that these generators obey the defining relations of the higher-rank Askey-Wilson algebra $AW(n)$. Equivalently, $AW(n)$, usually defined as the commutant of $U_q(\mathfrak{su}(1,1))$ in $U_q(\mathfrak{su}(1,1))^{\otimes n}$, is the same algebra as the commutant of $\mathfrak{o}_{q^{1/2}}(2)^{\oplus n}$ in this oscillator realization. The two descriptions are dual in the sense of Howe: the Casimir operators of the paired algebras are affinely related by $C_{i..j} = \frac{1}{(1+q)^2}\left(\Lambda_{[i;j]} - [j-i+1]_{q^{1/2}}[j-i-1]_{q^{1/2}}\right)$, which is Eq. (5.9). This affine pairing is what transfers the algebra structure from one picture to the other.

Load-bearing premise

The load-bearing assumption is that the explicit operators $\Lambda_i$ and $\Lambda_{ij}$ generate everything that commutes with the diagonal $\mathfrak{o}_{q^{1/2}}(2)^{\oplus n}$ subalgebra; the paper checks commutativity and linear independence but not completeness, and a larger commutant would break the identification with $AW(n)$.

Editorial extensions

If this is right

  • $AW(n)$ now has two explicit descriptions: as the commutant of $U_q(\mathfrak{su}(1,1))$ in its $n$-fold tensor product, and as the commutant of $\mathfrak{o}_{q^{1/2}}(2)^{\oplus n}$ in the $q$-oscillator representation of $\mathfrak{o}_{q^{1/2}}(2n)$.
  • The affine correspondence of Eq. (5.9) gives an explicit dictionary between the Casimir labels of $U_q(\mathfrak{su}(1,1))$ and the quadratic Casimirs of $\mathfrak{o}_{q^{1/2}}(2m)$.
  • In the $q\to1$ limit, the correspondence reduces to the known higher-rank Racah algebra result, so the $q$-deformed picture contains the classical one as a limit.
  • The $\Lambda$ operators provide a concrete oscillator model for $AW(n)$, expressing its generators directly in terms of $q$-oscillator creation and annihilation operators.
  • The paper identifies the $q\to-1$ limit as a route to the higher-rank Bannai-Ito algebra, leaving the explicit construction open.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension: the completeness of the commutant generation, asserted without proof even in the $n=3$ case, is a finite computation; checking it by direct linear algebra would either shore up or refute the identification with $AW(n)$.
  • Extension: if the oscillator picture is complete, imposing $\mathfrak{o}_{q^{1/2}}(2)^{\oplus n}$ invariance should yield new $q$-deformed superintegrable models by dimensional reduction, once a $q$-analogue of polar coordinates is found, which the paper notes is missing.
  • Extension: the affine pairing of Casimirs suggests that matrix elements of the $\Lambda$ operators in the oscillator basis should reproduce $q$-Racah or Askey-Wilson polynomial overlaps, giving a direct route to the polynomials themselves.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a second, Howe-dual realization of the Askey–Wilson algebra AW(3) as the commutant of o_{q^{1/2}}(2)⊕o_{q^{1/2}}(2)⊕o_{q^{1/2}}(2) inside q-oscillator realizations of o_{q^{1/2}}(6), and extends this picture to a higher-rank algebra AW(n) realized as the commutant of o_{q^{1/2}}(2)^{⊕n} in o_{q^{1/2}}(2n). The construction uses explicit quadratic-Casimir elements Λ_i, Λ_{ij} that commute with the relevant subalgebra, and affine relations (4.8) and (5.9) that identify these elements, up to shifts and scaling, with the intermediate Casimir elements of U_q(su(1,1)) arising in tensor-product embeddings. The paper argues that Howe duality for the pair (U_q(su(1,1)), o_{q^{1/2}}(2n)) from [26] explains this correspondence and yields the two dual descriptions of AW(n).

Significance. If the main assertions are correct, the paper provides a concrete and conceptually appealing new model of the Askey–Wilson algebra in terms of q-oscillators, complementing the standard tensor-product picture. The explicit affine correspondence between intermediate Casimirs of the two members of the dual pair is a valuable formula and generalizes the known classical Racah-algebra duality. A strength is that the derivations contain no parameter fitting and the q → 1 limit correctly matches existing results for the higher-rank Racah algebra. However, the paper's central claims currently rest on several unproved completeness and identification steps, so the significance can only be fully assessed after those gaps are closed.

major comments (4)
  1. [Section 5.2, Eqs. (5.2)–(5.3)] The assertion that the elements Λ_i and Λ_ij generate the full commutant of o_{q^{1/2}}(2)^{⊕n} is not proved. The text verifies only the commutation relations (5.3); no argument is given that these elements span the commutant, nor is a dimension count or an induction supplied for general n. For n=3, Section 4.1 states that six elements are independent but does not show that they exhaust the commutant. Since the identification of this commutant with AW(n) is the central claim, this missing completeness argument is load-bearing and must be supplied or replaced by a precise reference.
  2. [Section 4.1] The verification that K_A = \tildeΛ_12 and K_B = \tildeΛ_23 satisfy the AW(3) relations (2.3) is summarized as "a straightforward calculation" and is not shown. This is the key check for the n=3 result, and the reader cannot verify it from the text. The authors should include the explicit computation of the commutators and the resulting structure constants α, β, γ in terms of Λ_1, Λ_2, Λ_3 and Λ_123, or give a detailed outline sufficient for independent verification.
  3. [Section 5.2, Eq. (5.9)] The affine correspondence C_{i..j} = (1/(1+q)^2)(Λ_{[i;j]} − [j−i+1]_{q^{1/2}}[j−i−1]_{q^{1/2}}) is established only by asserting that β_{2m} is constant from a "quick look" and by evaluating α_{2m} on the ground state. This determines the constants at a single vector, not as an operator identity on the whole representation or on each joint irreducible component of the dual pair. To justify (5.9) one must show that Λ_{[i;j]} − (1+q)^2 C_{i..j} is a scalar operator, for example by proving that it is central and that the relevant representation is irreducible (or by computing its action on a basis). Without this, the correspondence could be an artifact of the ground-state evaluation.
  4. [Section 5.1 and Section 5.2] The paper acknowledges that the full set of relations of AW(n) is not known. Consequently, the statement that the Λ elements "realize AW(n)" is not checkable as an isomorphism of finitely presented algebras. If AW(n) is intended as the abstract algebra generated by the intermediate Casimirs C_A of U_q(su(1,1))^{⊗n}, the authors should state this explicitly and prove that the map C_A ↦ (1/(1+q)^2)(Λ_{[i;j]} − ...) extends to an injective algebra homomorphism whose image is the full commutant. As written, the equality of the two algebras is asserted rather than demonstrated, and the role of the incomplete presentation of AW(n) needs to be clarified.
minor comments (5)
  1. [Section 3.1, Eq. (3.2)] The notation q^{±1/4} would be clearer if the exponents were typeset explicitly as q^{±1/4}; the current display appears as q±1/4, which is ambiguous.
  2. [Section 4.1] The shorthand L122 and similar expressions should be consistently written as L_{12}^2 to avoid confusion with multiplication of the generator L_{12} by the scalar 2.
  3. [Section 5.2, Eq. (5.4)] The final sum in Eq. (5.4) has a garbled subscript: "Λ_{k−1+i, ℓ ≥ 3}" should presumably read "Λ_{k−1+i,k−1+i}" followed by the condition ℓ ≥ 3; please correct this typo.
  4. [Abstract and Section 1] There is a typo "the d ual pair" in the abstract; also "litterature" should be "literature" in Section 3.1.
  5. [Section 4.2, Eqs. (4.6)–(4.7)] The notation for the embedded generators J^ı_± uses a factor q^{A^0_{2i} + 1/2} in (4.6); it would be helpful to explicitly indicate how this matches the coproduct (2.6), since the J_0 shift by 1/4 per oscillator could lead to a phase ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the AW(n)-commutant identification rests on explicit oscillator computations and an external Howe-duality theorem, not on fitted parameters or load-bearing self-citation.

full rationale

The paper's central claim is that the commutant of o_{q^{1/2}}(2)^{oplus n} in the q-oscillator realization of o_{q^{1/2}}(2n) is generated by the Lambda_i and Lambda_ij and coincides with the higher-rank Askey-Wilson algebra AW(n). This is not obtained circularly. The Lambda elements are defined directly from the quadratic Casimir formula (3.3), and their commutativity with the subalgebra is verified in (5.3). The affine correspondence (5.9) between U_q(su(1,1)) intermediate Casimirs and the Lambda elements is derived from explicit oscillator expressions: the coefficient beta_{2m}=(1+q)^2 is read off from the coproduct and the shift alpha_{2m} is evaluated on the vacuum state. Neither constant is fitted to reproduce the AW(n) relations, and no target quantity is used as an input. The Howe-duality ingredient is the external theorem of Noumi-Umeda-Wakayama [26], not a self-citation; the authors' own prior work [24,25,39] appears only as classical or Bannai-Ito context and is not load-bearing. The genuine weakness is a proof gap: Section 5.2 states without proof that the Lambda elements 'generate its commutant', and Section 5.1 concedes that 'The full set of relations of AW(n) is not known'. These affect completeness and rigor, but they are not circular reductions because no definition or equation is shown to reduce to the target claim by construction. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the q-oscillator realization, the unproved generation statement for the commutant, and the imported Howe duality. There are no fitted parameters and no new postulated entities.

assumptions (4)
  • domain assumption The operators L_{i,i+1} defined in Eq. (3.9) satisfy the defining relations (3.1) of o_{q^{1/2}}(2n).
    Stated as a direct calculation in Section 3.3; no detailed proof is included.
  • domain assumption The elements Λ_i and Λ_{ij} generate the full commutant of o_{q^{1/2}}(2)^{⊕n} in the q-oscillator representation of o_{q^{1/2}}(2n).
    Asserted in Section 5.2 without proof; the paper checks commutativity and independence but not completeness.
  • domain assumption The higher rank Askey-Wilson algebra AW(n) is the algebra generated by the intermediate Casimir elements C_A of U_q(su(1,1)) in its n-fold tensor product, with relations as described in [35] and [36].
    Imported from prior literature; the paper notes in Section 5.1 that the full relations of AW(n) are not known.
  • domain assumption The pair (U_q(su(1,1)), o_{q^{1/2}}(2n)) forms a Howe dual pair with a multiplicity-free decomposition of the oscillator representation (1.1).
    Taken from [26], used in Sections 4.2 and 5.2 to justify the pairing of Casimirs. Not proven in this paper.

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Pith. "Pith review of The dual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(2n)\big)$, $q$-oscillators and Askey-Wilson algebras." pith.science (2026). https://pith.science/paper/5SYKXL7N

@misc{pith2026190804277,
  author       = {Pith},
  title        = {Pith review of: The dual pair $\big(U_q(\mathfraksu(1,1)),\mathfrako_q^1/2(2n)\big)$, $q$-oscillators and Askey-Wilson algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5SYKXL7N}},
  note         = {Machine review of arXiv:1908.04277}
}
abstract

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 $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(6)\big)$. We also generalize these results. A higher rank extension of the Askey-Wilson algebra denoted $AW(n)$ can be defined as the commutant of $U_q(\mathfrak{su}(1,1))$ in $U_q(\mathfrak{su}(1,1))^{\otimes n}$ and a dual description of $AW(n)$ as the commutant of $\mathfrak{o}_{q^{1/2}}(2)^{\oplus n}$ in $q$-oscillator representations of $\mathfrak{o}_{q^{1/2}}(2n)$ is offered by calling upon the dual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(2n)\big)$.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    math.QA 2019-08 conditional novelty 6.0 of 10

    New commutation and q-commutation relations are proven for generators of the higher rank Askey-Wilson and q-Bannai-Ito algebras, extending the rank-one defining relations.

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Works this paper leans on

39 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [26]

    Noumi, T

    M. Noumi, T. Umeda, and M. Wakayama. Dual pairs, spheric al harmonics and a Capelli identity in quantum group theory. Compos. Math. , 104(3):227–277, 1996

    work page 1996

  2. [1]

    A. S. Zhedanov. ”Hidden symmetry” of Askey-Wilson polyn omials. Theor. Math. Phys. , 89(2):1146–1157, 1991

    work page 1991

  3. [2]

    Baseilhac

    P. Baseilhac. Deformed Dolan-Grady relations in quantu m integrable models. Nucl. Phys. B, 709(3):491–521, 2005. arXiv:hep-th/0404149v3

    arXiv 2005

  4. [3]

    B. Aneva. Tridiagonal symmetries of models of nonequili brium physics. Symmetry, Integr. Geom. Methods Appl. , 4:056, 2008

    work page 2008

  5. [4]

    Vinet and A

    L. Vinet and A. Zhedanov. Quasi-linear algebras and inte grability (the Heisenberg pic- ture). Symmetry, Integr. Geom. Methods Appl. , 4:015, 2008

    work page 2008

  6. [5]

    Baseilhac

    P. Baseilhac. An integrable structure related with trid iagonal algebras. Nucl. Phys. B , 705(3):605–619, 2005. 12

    work page 2005

  7. [6]

    Baseilhac and K

    P. Baseilhac and K. Koizumi. A new (in)finite-dimensiona l algebra for quantum integrable models. Nucl. Phys. B , 720(3):325–347, 2005

    work page 2005

  8. [7]

    Terwilliger and R

    P. Terwilliger and R. Vidunas. Leonard Pairs and the Aske y-Wilson Relations. J. Algebr. Its Appl. , 03(04):411–426, 2005

    work page 2005

Show all 39 references
  1. [8]

    Terwilliger

    P. Terwilliger. The Universal Askey-Wilson Algebra. SIGMA. Symmetry, Integr. Geom. Methods Appl. , 7:069, 2011. arXiv:1104.2813

    2011 arXiv

  2. [9]

    Terwilliger

    P. Terwilliger. The Universal Askey-Wilson Algebra and DAHA of Type ( C ∨ 1 , C1). Sym- metry, Integr. Geom. Methods Appl. , 9:047, 2013

    2013

  3. [10]

    Terwilliger

    P. Terwilliger. The q-Onsager Algebra and the Universal Askey–Wilson Algebra. SIGMA. Symmetry, Integr. Geom. Methods Appl. , 14:044, 2018. arXiv:1801.06083

    2018 arXiv

  4. [11]

    Bullock and J

    D. Bullock and J. H. Przytycki. Multiplicative structu re of Kauffman bracket skein module quantizations. Proc. Am. Math. Soc. , 128(3):923–931, 1999. arXiv:math/9902117

    1999 arXiv

  5. [12]

    T. H. Koornwinder. The Relationship between Zhedanov’ s Algebra AW (3) and the Double Affine Hecke Algebra in the Rank One Case. SIGMA. Symmetry, Integr. Geom. Methods Appl., 3:063, dec 2006. arXiv:math/0612730

    2006 arXiv

  6. [13]

    T. H. Koornwinder. Zhedanov’s Algebra AW (3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra. Symmetry, Integr. Geom. Methods Appl., 4:052, 2008. arXiv:0711.2320v3

    2008 arXiv

  7. [14]

    Mazzocco

    M. Mazzocco. Confluences of the Painlev´ e equations, Ch erednik algebras and q-Askey scheme. Nonlinearity, 29(9):2565–2608, 2016

    2016

  8. [15]

    Y. A. Granovskii and A. Zhedanov. Hidden symmetry of the Racah and Clebsch-Gordan problems for the quantum algebra slq(2). J. Gr. Theory Phys. , 1:161–171, 1993

    1993

  9. [16]

    H. W. Huang. An embedding of the universal Askey–Wilson algebra into Uq(sl2)⊗Uq(sl2)⊗ Uq(sl2). Nucl. Phys. B , 922(3):401–434, 2017. arXiv:1611.02130

    2017 arXiv

  10. [17]

    R. Howe. Remarks on Classical Invariant Theory. Trans. Am. Math. Soc., 313(2):539–570, 1989

    1989

  11. [18]

    R. Howe. Transcending Classical Invariant Theory. J. Am. Math. Soc. , 2(3):535–552, 1989

    1989

  12. [19]

    R. Howe. Dual pairs in physics: harmonic oscillators, p hotons, electrons, singletons. In M. Flato, P. Sally, and G. Zuckerman, editors, Proceedings, Appl. Gr. Theory Phys. Math. Phys., chapter 6, pages 179–206. AMS, Chicago, 1987

    1987

  13. [20]

    Moshinsky and C

    M. Moshinsky and C. Quesne. Linear canonical transform ations and their unitary repre- sentations. J. Math. Phys. , 12(8):1772–1780, 1971

    1971

  14. [21]

    H. P. Jakobsen and M. Vergne. Wave and Dirac operators, a nd representations of the conformal group. J. Funct. Anal. , 24(1):52–106, 1977

    1977

  15. [22]

    D. J. Rowe, J. Repka, and M. J. Carvalho. Simple unified pr oofs of four duality theorems. J. Math. Phys. , 52(1), 2011. 13

    2011

  16. [23]

    D. J. Rowe, M. J. Carvalho, and J. Repka. Dual pairing of s ymmetry and dynamical groups in physics. Rev. Mod. Phys. , 84(2):711–757, 2012

    2012

  17. [24]

    Gaboriaud, L

    J. Gaboriaud, L. Vinet, S. Vinet, and A. Zhedanov. The Ra cah algebra as a commutant and Howe duality. J. Phys. A Math. Theor. , 51:50LT01, 2018

    2018

  18. [25]

    Gaboriaud, L

    J. Gaboriaud, L. Vinet, S. Vinet, and A. Zhedanov. The ge neralized Racah algebra as a commutant. J. Phys. Conf. Ser. , 1194:012034, 2019. arXiv:1808.09518v1

    2019 arXiv

  19. [27]

    Klimyk and K

    A. Klimyk and K. Schm¨ udgen. Quantum Groups and Their Representations . Springer- Verlag Berlin Heidelberg, 1997

    1997

  20. [28]

    A. M. Gavrilik and N. Z. Iorgov. On Casimir Elements of q-Algebras U ′ q(son) and Their Eigenvalues in Representations. Proc. Inst. Math. NAS Ukr. , 30(2):310–314, 2000

    2000

  21. [29]

    A. U. Klimyk. The nonstandard q-deformation of enveloping algebra U (son): results and problems. Czechoslov. J. Phys. , 51(4):331–340, 2001

    2001

  22. [30]

    A. U. Klimyk. On Classification of Irreducible Represen tations of q-Deformed Algebra U ′ q(son) Related to Quantum Gravity. Proc. Inst. Math. NAS Ukr. , 43(2):407–418, 2002. arXiv:math/0110038

    2002 arXiv

  23. [31]

    N. Z. Iorgov and A. U. Klimyk. Classification Theorem on I rreducible Representations of the q-deformed algebra U ′ q(son). Int. J. Math. Math. Sci. , 2:225–262, 2005

    2005

  24. [32]

    M. Noumi. Macdonald’s Symmetric Polynomials as Zonal S pherical Functions on Some Quantum Homogeneous Spaces. Adv. Math. (N. Y). , 123(1):16–77, oct 1996

    1996

  25. [33]

    Feigin and T

    M. Feigin and T. Hakobyan. On Dunkl angular momenta alge bra. J. High Energy Phys. , 2015(11):107, 2015

    2015

  26. [34]

    De Bie, V

    H. De Bie, V. X. Genest, W. van de Vijver, and L. Vinet. A hi gher rank Racah algebra and the Zn 2 Laplace–Dunkl operator. J. Phys. A Math. Theor. , 51(2):025203, 2017

    2017

  27. [35]

    Post and A

    S. Post and A. Walter. A higher rank extension of the Aske y-Wilson Algebra. 2017. arXiv:1705.01860v2

    2017 arXiv

  28. [36]

    De Bie, H

    H. De Bie, H. De Clercq, and W. Van De Vijver. The higher ra nk q-deformed Bannai-Ito and Askey-Wilson algebra. 2018. arXiv:1805.06642v1

    2018 arXiv

  29. [37]

    E. G. Kalnins, W. Miller, and S. Post. Wilson polynomial s and the generic superintegrable system on the 2-sphere. J. Phys. A Math. Theor. , 40(38):11525–11538, sep 2007

    2007

  30. [38]

    P. Iliev. The generic quantum superintegrable system o n the sphere and Racah operators. Lett. Math. Phys. , 107(11):2029–2045, nov 2017

    2017

  31. [39]

    Gaboriaud, L

    J. Gaboriaud, L. Vinet, S. Vinet, and A. Zhedanov. The du al pair P in(2n) ⊗ osp(1|2), the Dirac equation and the Bannai–Ito algebra. Nucl. Phys. B , 937:226–239, 2018. 14

    2018

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