# Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of $U_q(\mathfrak{sl}_2)$

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.
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-02933969
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Submitted on : Tuesday, September 8, 2020 - 11:01:06 PM
Last modification on : Wednesday, September 23, 2020 - 3:02:26 AM

### Citation

Nicolas Crampé, Luc Vinet, Meri Zaimi. Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of $U_q(\mathfrak{sl}_2)$. 2020. ⟨hal-02933969⟩

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