On Endomorphisms of Quantum Tensor Space |
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Authors: | Gustav Isaac Lehrer Ruibin Zhang |
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Institution: | 1. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia
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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. |
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