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

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Abstract

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.

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Acknowledgements

The authors are grateful to Loïc Poulain D’Andecy for numerous enlightening discussions. They also thank Paul Terwilliger for useful exchanges. N. Crampé is partially supported by Agence Nationale de la Recherche Projet AHA ANR-18-CE40-0001. The work of L. Vinet is funded in part by a discovery grant of the Natural Sciences and Engineering Research Council (NSERC) of Canada. M. Zaimi holds graduate scholarships from the NSERC and the Fonds de recherche du Québec—Nature et technologies (FRQNT).

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Correspondence to Meri Zaimi.

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Communicated by Nikolai Kitanine.

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Crampé, N., Vinet, L. & Zaimi, M. Temperley–Lieb, Birman–Murakami–Wenzl and Askey–Wilson Algebras and Other Centralizers of \(U_q(\mathfrak {sl}_2)\). Ann. Henri Poincaré 22, 3499–3528 (2021). https://doi.org/10.1007/s00023-021-01064-x

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