A categorical framework for glider representations

Abstract

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 $\Gamma$-filtered ring $FR$ and a subset $\Lambda \subseteq \Gamma$, we provide a category $\operatorname{Glid}_\Lambda FR$ 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 $\operatorname{Glid}_\Lambda FR$ 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 $k \cdot 1 \subseteq R$ of a bialgebra $R$ is sufficient to recover the bialgebra $R$ by recovering the usual fiber functor from $\operatorname{Glid}_\Lambda FR.$ When applied to a group algebra $kG$, this shows that the monoidal category $\operatorname{Glid}_\Lambda F(kG)$ alone is sufficient to distinguish even isocategorical groups.