2020/03/12 by Ruben Henrard, Henrard, Ruben, Adam-Christiaan van Roosmalen +1
Mathematics · #16W70 #18E10 #18E35 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:16W70 #msc:18E10 #msc:18E35
paper · pdf · doi:10.48550/arxiv.2003.05930
33 pages, comments welcome
arxiv created 2020/03/12 · arxiv updated 2020/03/13
Fragment and glider representations (introduced by F. Caenepeel, S. Nawal, and F. Van Oystaeyen) form a generalization of filtered modules over a filtered ring. Given a Γ-filtered ring FR and a subset Λ⊆ Γ, we provide a category GlidΛFR of glider representations, and show that it is a complete and cocomplete deflation quasi-abelian category. We discuss its derived category, and its subcategories of natural gliders and Noetherian gliders. If R is a bialgebra over a field k and FR is a filtration by bialgebras, we show that GlidΛFR is a monoidal category which is derived equivalent to the category of representations of a semi-Hopf category (in the sense of E. Batista, S. Caenepeel, and J. Vercruysse). We show that the monoidal category of glider representations associated to the one-step filtration k ⋅ 1 ⊆ R of a bialgebra R is sufficient to recover the bialgebra R by recovering the usual fiber functor from GlidΛFR. When applied to a group algebra kG, this shows that the monoidal category GlidΛF(kG) alone is sufficient to distinguish even isocategorical groups.