The Birman-Wenzl-Murakami algebra, Hecke algebra and representations of U_{q}(osp(1|2n))

Abstract

A representation of the Birman-Wenzl-Murakami algebra BW_{t}(-q^{2n},q) exists in the centraliser algebra End_{U_q(osp(1|2n))}(V^{\otimes t}), where V is the fundamental (2n+1)-dimensional irreducible U_{q}(osp(1|2n))-module. This representation is defined using permuted R-matrices acting on V^{\otimes t}. A complete set of projections onto and intertwiners between irreducible U_{q}(osp(1|2n))-summands of V^{\otimes t} exists via this representation, proving that End_{U_q(osp(1|2n))}V^{\otimes t}) is generated by the set of permuting R-matrices acting on V^{\otimes t}. We also show that a representation of the the Iwahori-Hecke algebra H_{t}(-q) of type A_{t-1} exists in the centraliser algebra End_{U_q(osp(1|2))}[(V^{+}_{1/2})^{\otimes t}], where V^{+}_{1/2} is a two-dimensional irreducible representation of U_{q}(osp(1|2)).