On tensor spaces for Birman–Murakami–Wenzl algebras

Let l,n∈N. Let sp2l be the symplectic Lie algebra over the complex number field C. Let V be the natural representation of the quantized enveloping algebra Uq(sp2l) and Bn,q the specialized Birman–Murakami–Wenzl algebra with parameters −q2l+1,q. In this paper, we construct a certain element in the annihilator of V⊗n in Bn,q, which comes from some one-dimensional two-sided ideal of Birman–Murakami–Wenzl algebra and is explicitly characterized (modulo the determination of some constants). We prove that the two-sided ideal generated by this element is indeed the whole annihilator of V⊗n in Bn,q and conjecture that the same is true over arbitrary ground fields and for any specialization of the parameter q. The conjecture is verified in the case when q is specialized to 1 (i.e., the Brauer algebra case) and the case when n=l+1.