On Endomorphisms of Quantum Tensor Space

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We give a presentation of the endomorphism algebra End𝓊q(sl2)(

), where V is the three-dimensional irreducible module for quantum sl2 over the function field ℂ(q½). 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 𝓊q(sl2)-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

are consequences of relations among the three R-matrices acting on

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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