Abstract
In this paper, we prove that for any pair of weak Hopf monoids H and B in a symmetric monoidal category where every idempotent morphism splits, the category of H-B-Long dimodules HBLong is monoidal. Moreover, if H is quasitriangular and B coquasitriangular, we also prove that HBLong is braided. As a consequence of this result, we obtain that if H is triangular and B cotriangular, HBLong is an example of a symmetric monoidal category.
Highlights
Let R be a commutative fixed ring with unit and let C be the non-strict symmetric monoidal category of R-Mod where ⊗ denotes the tensor product over R
For two weak Hopf monoids H and B, in the third section we introduce the category of H-B-Long dimodules, denoted as for the category R-Mod, by BH Long and we describe in detail the tensor product of this category
We have proven that if H and B are weak Hopf algebras in a symmetric monoidal category where every idempotent morphism splits, the category of H-B-Long dimodules, denoted by BH Long, is monoidal
Summary
Let R be a commutative fixed ring with unit and let C be the non-strict symmetric monoidal category of R-Mod where ⊗ denotes the tensor product over R. In [4] (see [5]) we can find the definition of Long dimodule for Hopf quasigroups and, if H is a quasitriangular Hopf quasigroup and B coquasitriangular Hopf quasigroup, as in the previous settings, the authors prove that the category of left-left H-B-Long dimodules is a braided monoidal subcategory of the category of Yetter-Drinfel’d modules over the Hopf quasigroup H ⊗ B. For two weak Hopf monoids H and B, in the third section we introduce the category of H-B-Long dimodules, denoted as for the category R-Mod, by BH Long and we describe in detail the tensor product of this category In this setting the tensor product is defined as the image of the composition of two idempotent morphisms associated with the module and comodule structure, respectively. If H is triangular and B cotriangular, we established that BH Long is symmetric
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
