Letters in Mathematical Physics

, Volume 86, Issue 2–3, pp 209–227 | Cite as

On Endomorphisms of Quantum Tensor Space

  • Gustav Isaac LehrerEmail author
  • Ruibin Zhang


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.

Mathematics Subject Classification (2000)

Primary 17B10 17B37 Secondary 20C08 


cellular algebras Lie algebras quantum groups presentations of endomorphism algebras BMW algebra Brauer algebra 


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© Springer 2008

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

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