We construct analogs of the embedding of orthogonal and symplectic groups into unitary groups in the context of fusion categories. At least some of the resulting module categories also appear in boundary conformal field theory. We determine when thes
e categories are unitarizable, and explicitly calculate the index and principal graph of the resulting subfactors.
We give a presentation of the centralizer algebras for tensor products of spinor representations of quantum groups via generators and relations. In the even-dimensional case, this can be described in terms of non-standard q-deformations of orthogonal
Lie algebras; in the odd-dimensional case only a certain subalgebra will appear. In the classical case q = 1 the relations boil down to Lie algebra relations.
We define a new $q$-deformation of Brauers centralizer algebra which contains Hecke algebras of type $A$ as unital subalgebras. We determine its generic structure as well as the structure of certain semisimple quotients. This is expected to have appl
ications for constructions of subfactors of type II$_1$ factors and for module categories of fusion categories of type $A$ corresponding to certain symmetric spaces.
