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.

26 pages

References in corpus (3)

• The Universal Askey-Wilson Algebra
• A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$
• Fused braids and centralisers of tensor representations of $U_q(g{l}_N)$

Cited by in corpus (5)

• Racah algebras, the centralizer $Z_n({\mathfrak{sl}}_2)$ and its Hilbert-Poincaré series
• Askey-Wilson braid algebra and centralizer of $U_q({\mathfrak{sl}}_2)$
• Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors
• Braid group and $q$-Racah polynomials
• Chern-Simons theory, link invariants and the Askey-Wilson algebra