On endomorphisms of quantum tensor space

Abstract

We give a presentation of the endomorphism algebra $\End_{\cU_q(\fsl_2)}(V^{\otimes r})$, where $V$ is the 3-dimensional irreducible module for quantum $\fsl_2$ over the function field $\C(q^{{1/2}})$. This will be as a quotient of the Birman-Wenzl-Murakami algebra $BMW_r(q):=BMW_r(q^{-4},q^2-q^{-2})$ by an ideal generated by a single idempotent $\Phi_q$. Our presentation is in analogy with the case where $V$ is replaced by the 2- dimensional irreducible $\cU_q(\fsl_2)$-module, the BMW algebra is replaced by the Hecke algebra $H_r(q)$ of type $A_{r-1}$, $\Phi_q$ is replaced by the quantum alternator in $H_3(q)$, and the endomorphism algebra is the classical realisation of the Temperley-Lieb algebra on tensor space. In particular, we show that all relations among the endomorphisms defined by the $R$-matrices on $V^{\otimes r}$ are consequences of relations among the three $R$-matrices acting on $V^{\otimes 4}$. The proof makes extensive use of the theory of cellular algebras. Potential applications include the decomposition of tensor powers when $q$ is a root of unity.