In this paper, the Drinfeld center of a monoidal category is generalized to a class of mixed Drinfeld centers. This gives a unified picture for the Drinfeld center and a natural Heisenberg analogue. Further, there is an action of the former on the latter. This picture is translated to a description in terms of Yetter-Drinfeld and Hopf modules over quasi-bialgebras in a braided monoidal category. Via braided reconstruction theory, intrinsic definitions of braided Drinfeld and Heisenberg doubles are obtained, together with a generalization of the result of Lu [22] that the Heisenberg double is a 2-cocycle twist of the Drinfeld double for general braided Hopf algebras.

We explicitly compute a monoidal subcategory of the monoidal center of Deligne's interpolation category normalRe̲pfalse(Stfalse), for t not necessarily a natural number, and we show that this subcategory is a ribbon category. For t=n, a natural number, there exists a functor onto the braided monoidal category of modules over the Drinfeld double of Sn which is essentially surjective and full. Hence the new ribbon categories interpolate the categories of crossed modules over the symmetric groups. As an application, we obtain invariants of framed ribbon links which are polynomials in the interpolating variable t. These polynomials interpolate untwisted Dijkgraaf–Witten invariants of the symmetric groups.

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.

This paper develops a theory of monoidal categories relative to a braided monoidal category, called augmented monoidal categories. For such categories, balanced bimodules are defined using the formalism of balanced functors. The two main constructions are a relative tensor product of monoidal categories as well as a relative version of the monoidal center, which are Morita dual constructions. A general existence statement for a relative tensor products is derived from the existence of pseudo-colimits.In examples, a category of locally finite weight modules over a quantized enveloping algebra is equivalent to the relative monoidal center of modules over its Borel part. A similar result holds for small quantum groups, without restricting to locally finite weight modules. More generally, for modules over braided bialgebras, the relative center is shown to be equivalent to the category of braided Yetter-Drinfeld modules (or crossed modules). This category corresponds to modules over the braided Drinfeld double (or double bosonization) which are locally finite for the action of the dual.1.3. Summary of Results. In the spirit of moving from the representation theory of algebras to modules of categories (in this case, monoidal categories), this paper studies the representation theory of the relative monoidal center in the framework of [EGNO15]. After establishing the categorical setup and recalling generalities on categorical modules (Sections 2.1 & 3.1), the relative tensor product of categorical bimodules is reviewed in the generality of finitely cocomplete k-linear categories (Section 3.2 & Appendix A).The concept of the relative monoidal center from [Lau15] is refined using the language of Bbalanced functors to allow a better study of its categorical modules. For this, B-augmented monoidal categories are introduced in Section 3.3. These can be thought of as categorical analogues of Caugmented C-algebras R over a commutative k-algebra C. Over a B-augmented monoidal category, we can study B-balanced bimodules (Section 3.4). This construction is a categorical analogue of R b C R op -modules, over a C-algebra R. In other words, R-bimodules for which the left and right C-action coincide. Based on this concept, we present two constructions for a B-augmented monoidal category C:(1) the relative monoidal center Z B pCq, which is defined as Z B pCq " Hom B C-C pC, Cq, i.e. the category of endofunctors of B-balanced bimodule functors of the regular C-bimodule (Section 3.5);(2) the relative tensor product C B C op (Theorem 3.21) which is a monoidal category, generalizing the tensor product of categories of Kelly [Kel05]. The monoidal category C B C op has the universal property that its categorical modules give all B-balanced bimodules over C (Theorem 3.27). We show that there is a natural construction to turn a C-bimodule into a B-balanced C-bimodule.Theorem 3.42. Restriction along the monoidal functor C C op Ñ C B C op has a left 2-adjointThis construction is a categorical analogue of restricting a R-R-bimodule to the subspa...