Fusion symmetric spaces and subfactors

Abstract. 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 these cate-gories are unitarizable, and explicitly calculate the index and principal graph of the resulting subfactors. This paper is a sequel of our previous paper [W4], where we introduced a q-deformation of Brauer’s centralizer algebra for orthogonal and symplectic groups; this algebra had already appeared more or less before in [Mo], see also discussion in [W4]. It is motivated by finding a deformation of orthogonal or symplectic subgroups of a unitary group which is compatible with the standard quantum deformation of the big group. This has been done before on the level of coideal subalgebras of Hopf algebras by Letzter. However, our categorical approach also allows us to extend this to the level of fusion tensor categories, where we find finite analogs of symmetric spaces related to the already mentioned groups. Moreover, we can establish C∗ structures, necessary for the construction of subfactors, in this categorical setting; this is not so obvious to see in the setting of co-ideal algebras. It is well-known how one can use a subgroup H of a (for simplicity here) finite group