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

Abstract

The q-deformed Bannai-Ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra osp q (1|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 A q n , 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 osp q (1|2). An explicit realization in terms of q-shift operators and reflections is proposed, which will be called the Z n 2 q-Dirac-Dunkl model. The algebra A q n is shown to arise as the symmetry algebra of the constructed Z n 2 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 CK-extension and Fischer decomposition.

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

“…Such higher rank algebras are motivated by their role as symmetry algebras for superintegrable quantum systems of higher dimension. This has been confirmed in the limiting case q = 1 [11,9,10], and later also for general q [8]. In both cases, the Hamiltonians under consideration are built from Dunkl operators with Z n 2 symmetry [12], possibly q-deformed [5].…”

“…The construction of AW (n) is rather intricate: in [8] we have outlined an algorithm which repeatedly applies the U q (sl 2 )-coproduct ∆ and a coaction τ to the Casimir element Λ, in a specific order. This way we construct, as an extension of (2)-(3)-(4), an element Λ A ∈ U q (sl 2 ) ⊗n for each A ⊆ {1, .…”

“…In [8] this approach was generalized to n-fold tensor products for arbitrary n, which leads to an extension of the Askey-Wilson algebra to higher rank, denoted AW (n). The same algorithm allows to construct a higher rank extension for the q-Bannai-Ito algebra, which is isomorphic to AW (3) under a transformation q → −q 2 and allows a similar embedding in osp q (1|2) ⊗3 [15].…”

Bannai–Ito algebras and the universal R-matrix of $$\pmb {\mathfrak {osp}}(1|2)$$

“…This universal R-matrix approach has already been applied to the study of the Askey-Wilson algebra AW (3) [10] as the centralizer of the diagonal action of U q (sl(2)) into its threefold product and has also been seen to hold promises for advancing the understanding of the higher rank AW (n) where one is looking at the centralizer of U q (sl(2)) in U q (sl(2)) ⊗n [2]. While advances have been made on this last front [3], [14], a complete description of AW (n) is still lacking. We trust that the treatment given here of the Bannai-Ito algebra BI(n) using the universal R -matrix might hold the clues towards bringing this quest to a satisfactory conclusion.…”

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

“…• The Askey-Wilson algebra admits an embedding into U q (sl 2 ) ⊗ U q (sl 2 ) ⊗ U q (sl 2 ) [GZ93b] (see also [H16]), where the generators map as A → ∆(C) ⊗ 1, A * → 1 ⊗ ∆(C) with C, ∆, respectively the Casimir element and coproduct of U q (sl 2 ). In the recent literature [PW17,DDV18], a generalization of the Askey-Wilson algebra indexed by N has been introduced. For N = 3, it reduces to the Askey-Wilson algebra.…”