Letters in Mathematical Physics

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

On Endomorphisms of Quantum Tensor Space

Article

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 BMWr(q) : =  BMWr(q−4, q2 − q−2) by an ideal generated by a single idempotent Φq. 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 Hr(q) of type Ar-1, Φq is replaced by the quantum alternator in H3(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 

Keywords

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

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Copyright information

© Springer 2008

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

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