Constructing Non-semisimple Modular Categories with Local Modules

Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras

“…The category of local modules Rep loc Z(Vect ω G ) (A) over a rigid Frobenius algebra A as in Section 1.2 is a modular category by [28,Theorem 4.12] and [26,Theorem 4.5] in the semisimple case. In fact, [9,Theorem 3.16] shows that such modular categories are equivalent as ribbon categories to Z Vect ω H/N for a 3-cocycle ω on H/N such that its pullback to H via the quotient homomorphism is equivalent to ω| H .…”

“…In the following, we recall local modules over commutative algebras in a braided category C [26,28,37,40]. Definition 2.11.…”

“…Then Schauenburg's result gives an equivalence of braided monoidal categories between Z(Rep C (A + )) and Rep loc Z(C) (A + ). Categories of local modules are a source of modular tensor categories, both in the semisimple case [26,Theorem 4.5] and the non-semisimple case [28,Theorem 4.12]. For this, recall that a rigid Frobenius algebra in a ribbon category C is a connected commutative algebra satisfying the equivalent conditions of Proposition 2.5.…”

“…Its category of representations will be then equivalent to Z Vect ω G [32]. Given a commutative algebra A in a braided tensor category C one defines a braided tensor category Rep loc C (A) of local modules [26,28,37,40]. Such categories of local modules have been of particular interest in the mathematical physics literature, see, e.g., [17,18].…”

“…For instance, categories of local modules relate the representations of a VOA to those of its extensions [6,23,26]. Given a rigid Frobenius algebra (i.e., a connected commutative special Frobenius algebra) in C, it was shown that the rigid monoidal category Rep loc C (A) of local modules is again modular (see [26] in the semisimple case, and [28] in the general case). Such rigid Frobenius algebras were classified for the semisimplification of U q (sl 2 )-modules [26], for the Drinfeld center of modules over a finite group [7,28], and for DW categories Z Vect ω kG in char k = 0 [9].…”