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 categories 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 G to construct a module category of the representation category Rep G of G. This module category also appears in the context of subfactors of II1 von Neumann factors as follows: Let R be the hyperfinite II1 factor, and let N = RG ⊂M = RH be the fixed points under outer actions of G and H. Then the category of N − N bimodules is equivalent to Rep G, and the module category is given via the M−N bimodules of the inclusion N ⊂ M; its simple objects are labeled by the irreducible representations of H. In particular, an important invariant called the principal graph of the subfactor is determined by the restriction rules for representations from G to H. Important examples of subfactors were constructed from fusion categories whose Grothendieck semirings are quotients of the ones of semisimple Lie groups. So a natural question to ask is whether one can perform a similar construction in this context. More precisely, can we find restriction rules for type A fusion categories which describe a subfactor as before, and which will approach in the classical limit the usual restriction rules from U(N) to O(N). We answer this question in the positive in this paper via a fairly elementary construction. We show that certain semisimple quotients of the q-Brauer algebras have a C∗ structure and contain C∗-quotients of Hecke algebras of type A. The subfactor is then obtained as the closure of inductive limits of such algebras. Due to its close connection to Lie groups, we can