On R-Matrix Representations of Birman–Murakami–Wenzl Algebras

It is shown that, to every local representation of the Birman–Murakami–Wenzl algebra defined by a skew invertible R-matrix $\hat R\in\mathrm{Aut}(V^{\otimes 2})$, one can associate pairings $V\otimes V\rightarrow\mathbb C$ and $V^*\otimes V^*\rightarrow\mathbb C$, where $V$ is the representation space. Further, conditions are investigated under which the corresponding quantum group is of SO or Sp type.