Frobenius algebras in tensor categories and bimodule extensions

By recent research developments, the notion of tensor category has been recognized as a fundamental language in describing quantum symmetry, which can replace the traditional method of groups for investigating symmetry. The terminology of tensor category is used here as a synonym of linear monoidal category and hence it has a good affinity with semigroup. One way to incorporate the invertibility axiom of groups is to impose rigidity (or duality) on tensor categories, which will be our main standpoint in what follows. When a tensor category bears a finite group symmetry inside, it is an interesting problem to produce a new tensor category by taking quotients with respect to this inner symmetry. For quantum symmetries of rational conformal field theory, this kind of constructions are worked out in a direct and individual way with respect to finite cyclic groups. In our previous works, these specific constructions are organized by interpreting them as bimodule tensor categories for the symmetry of finite groups with a satisfactory duality on bimodule extensions [12]. The construction is afterward generalized to the symmetry of tensor categories governed by finite-dimensional Hopf algebras [13]. We shall present in this paper a further generalization to symmetries described by categorical Frobenius algebras, which are formulated and utilized by J. Fuchs and C. Schweigert for a mathematical description of boundary conditions in conformal field theory [3] (see [5] for earlier studies on categorical Frobenius structures). A similar notion has been introduced under the name of Q-systems by R. Longo in connection with subfactory theory ([6], cf. also [9]). More precisely, a Q-system, if it is algebraically formulated, is equivalent to giving a Frobenius algebra satisfying a certain splitting condition, which is referred to as a special Frobenius algebra according to the terminology in [3].