The endomorphism algebras for the action of the quantum symplectic and orthogonal groups on the $k$-fold tensor product of the respective natural representations are generated by the action Birman-Murakami-Wenzl algebras.

We construct the irreducible representations of the Birman-Murakami-Wenzl algebras, using a cellular basis in the sense of Graham and Lehrer.

Via an appropriate specialisation, these results will also hold for the Brauer algebras.