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.
The higher rank Askey-Wilson algebra was recently constructed in the n-fold tensor product of Uq(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. µ {2,4,5,8} 2 µ {2,4,5,8} 3 (Λ) ⊗ 1, with µ {2,4,5,8} 3Again, the idea is that each element of A but the maximum a m corresponds to an application of ∆, whereas τ L fills up the holes. However, as opposed to Definition 2.3, we now run through the elements of A in decreasing order, from right to left. This is why we refer to the algorithm of Definition 2.5 as the left extension process.
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.
Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra g, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras by onesided coideal subalgebras of the quantized enveloping algebra Uq(g). We provide a complete presentation by generators and relations for these quantum symmetric pair coideal subalgebras. These relations are of inhomogeneous q-Serre type and are valid without restrictions on the generalized Cartan matrix. We draw special attention to the split case, where the quantum symmetric pair coideal subalgebras are generalized q-Onsager algebras.
This paper is the first in a series on graphical calculus for quantum vertex operators. We establish in great detail the foundations of graphical calculus for ribbon categories and braided monoidal categories with twist. We illustrate the potential of this approach by applying it to various categories of quantum group modules, in particular to derive an extension of the linear operator equation for dynamical fusion operators, due to Arnaudon, Buffenoir, Ragoucy and Roche, to a system of linear operator equations of q-KZ type.Contents 1.5. The ribbon element and twisting in category M 1.6. Dual representations 2. Graphical calculus 2.1. Ribbon graphs and ribbon graph diagrams 2.2. Ribbon-braid graphs and ribbon-braid graph diagrams 2.3. Strictifications 2.4. Graphical calculus for D-colored ribbon-braid graphs 2.5. The Reshetikhin-Turaev functor 2.6. Colored ribbon graph subdiagrams in B D 2.7. Fusion morphisms 2.8. Bundling of strands 3. q-KZ equations for k-point dynamical fusion operators 3.1. k-point quantum vertex operators and their graphical representation 3.2. The topological operator q-KZ equations 3.3. The topological q-KZ equations for the k-point dynamical fusion operator Conclusion References
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