On Endomorphisms of Quantum Tensor Space
نویسنده
چکیده
We give a presentation of the endomorphism algebra EndUq(sl2)(V ), where V is the 3-dimensional irreducible module for quantum sl2 over the function field C(q 1 2 ). This will be as a quotient of the Birman-Wenzl-Murakami algebra BMWr(q) := BMWr(q , q2− q) by an ideal generated by a single idempotent Φq. Our presentation is in analogy with the case where V is replaced by the 2dimensional 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 TemperleyLieb algebra on tensor space. In particular, we show that all relations among the endomorphisms defined by the R-matrices on V ⊗r are consequences of relations among the three R-matrices acting on V . 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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