On Endomorphisms of Quantum Tensor Space

Abstract

We give a presentation of the endomorphism algebra $${\rm End}_{\mathcal {U}_{q}(\mathfrak {sl}_{2})}(V^{\otimes r})$$ , where V is the three-dimensional irreducible module for quantum $${\mathfrak {sl}_2}$$ over the function field $${\mathbb {C}(q^{\frac{1}{2}})}$$ . 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 Φ q . Our presentation is in analogy with the case where V is replaced by the two-dimensional irreducible $${\mathcal {U}_q(\mathfrak {sl}_{2})}$$ -module, the BMW algebra is replaced by the Hecke algebra H r (q) of type A r-1, Φ 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.