2018/03/31 by Robert Laugwitz · 8 citations
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Center (category theory) #Closed monoidal category #Enriched category #Homotopy and Cohomology in Algebraic Topology #Hopf algebra #Monoidal category #Symmetric monoidal category #Tensor product
paper · open access · doi:10.1142/s0219199719500688
published in Communications in Contemporary Mathematics 22(08), 1950068 (World Scientific)
openalex created_date 2018/03/29 · openalex publication_date 2019/08/13 · crossref created 2019/08/13 · crossref issued 2019/09/19 · crossref published 2019/09/19 · crossref published-online 2019/09/19 · crossref deposited 2020/08/31 · crossref published-print 2020/12/01 · crossref indexed 2026/07/31 · openalex updated_date 2026/08/05
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. It is shown that there exists a monoidal structure on the relative tensor product of two augmented monoidal categories which is Morita dual to a relative version of the monoidal center. 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 bialgebras inside a braided monoidal category, the relative center is shown to be equivalent to the category of Yetter–Drinfeld modules inside the braided category. If the braided category is given by modules over a quasitriangular Hopf algebra, then the relative center corresponds to modules over a braided version of the Drinfeld double (i.e. the double bosonization in the sense of Majid) which are locally finite for the action of the dual.