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

Clercq, Hadewijch De

doi:10.3842/sigma.2019.099

Cited by 10 publications

(27 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…are easily checked to coincide with (23), (24) and ( 25) by ( 5) and (19). The equation for Γ q [1;2] then follows immediately from (14). Suppose now the claim holds for n − 1.…”

Section: 3mentioning

confidence: 87%

See 1 more Smart Citation

The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials

De Bie,

De Clercq

2019

Preprint

Self Cite

View full text Add to dashboard Cite

The Gasper and Rahman multivariate (−q)-Racah polynomials appear as connection coefficients between bases diagonalizing different abelian subalgebras of the recently defined higher rank q-Bannai-Ito algebra A q n . Lifting the action of the algebra to the connection coefficients, we find a realization of A q n by means of difference operators. This provides an algebraic interpretation for the bispectrality of the multivariate (−q)-Racah polynomials, as was established in Iliev, Trans. Amer. Math. Soc., 363(3) (2011), 1577-1598.Furthermore, we extend the Bannai-Ito orthogonal polynomials to multiple variables and use these to express the connection coefficients for the q = 1 higher rank Bannai-Ito algebra An, thereby proving a conjecture from De Bie et al., Adv. Math. 303 (2016), 390-414. We derive the orthogonality relation of these multivariate Bannai-Ito polynomials and provide a discrete realization for An.

show abstract

“…are easily checked to coincide with (23), (24) and ( 25) by ( 5) and (19). The equation for Γ q [1;2] then follows immediately from (14). Suppose now the claim holds for n − 1.…”

Section: 3mentioning

confidence: 87%

“…The proof was given for sets of consecutive integers in [7], which is the only case we will rely on in this paper. For a proof of the general case we refer to [14].…”

Section: The Higher Rank Q-bannai-ito Algebramentioning

confidence: 99%

The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials

De Bie,

De Clercq

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…[27] for a review). Many attempts [44][45][46] to generalize this result to n-fold tensor products have yielded relations of this centralizer but certainly did not give all the defining relations. Looking ahead, we are planning to provide a complete set of defining relations, by using some deformation of the defining relations of the special Racah algebra given in this paper.…”

Section: Discussionmentioning

confidence: 99%

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

Crampe,

Gaboriaud,

d'Andecy

et al. 2021

Preprint

View full text Add to dashboard Cite

The higher rank Racah algebra R(n) introduced in [1] 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 (sl 2 ) of the diagonal embedding of U (sl 2 ) in U (sl 2 ) ⊗n . This leads to a first and novel presentation of the centralizer Z n (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.

show abstract

“…These will lead us to a realization of the higher rank

q

‐Bannai–Ito algebra

A_{n}^{q}

, or more precisely of the algebra abstractly defined by the relations of Proposition 1. At this time, it is not clear whether the two are isomorphic, we refer to [14] for a discussion. Nevertheless, we will take the freedom to call the operator system we are about to construct a realization of the algebra

A_{n}^{q}

.…”

Section: Realization With Difference Operatorsmentioning

confidence: 99%

The q‐Bannai–Ito algebra and multivariate (−q)‐Racah and Bannai–Ito polynomials

Bie

Clercq

2020

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

The Gasper and Rahman multivariate (−q)‐Racah polynomials appear as connection coefficients between bases diagonalizing different abelian subalgebras of the recently defined higher rank q‐Bannai–Ito algebra Anq. Lifting the action of the algebra to the connection coefficients, we find a realization of Anq by means of difference operators. This provides an algebraic interpretation for the bispectrality of the multivariate (−q)‐Racah polynomials, as was established in Iliev (Trans. Amer. Math. Soc. 363(3) (2011) 1577–1598). Furthermore, we extend the Bannai–Ito orthogonal polynomials to multiple variables and use these to express the connection coefficients for the q=1 higher rank Bannai–Ito algebra An, thereby proving a conjecture from De Bie et al. (Adv. Math. 303 (2016) 390–414). We derive the orthogonality relation of these multivariate Bannai–Ito polynomials and provide a discrete realization for An.

show abstract

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

Cited by 10 publications

References 36 publications

The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials

The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials

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

The q‐Bannai–Ito algebra and multivariate (−q)‐Racah and Bannai–Ito polynomials

Contact Info

Product

Resources

About