Temperley–Lieb, Brauer and Racah algebras and other centralizers of 𝔰𝔲(2)

Abstract

In the spirit of the Schur–Weyl duality, we study the connections between the Racah algebra and the centralizers of tensor products of three (possibly different) irreducible representations of s u ( 2 ) \mathfrak {su}(2) . As a first step we show that the Racah algebra always surjects onto the centralizer. We then offer a conjecture regarding the description of the kernel of the map, which depends on the irreducible representations. If true, this conjecture would provide a presentation of the centralizer as a quotient of the Racah algebra. We prove this conjecture in several cases. In particular, while doing so, we explicitly obtain the Temperley–Lieb algebra, the Brauer algebra and the one-boundary Temperley–Lieb algebra as quotients of the Racah algebra.