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Register a new userA categorical framework for glider representations
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Adam-Christiaan van Roosmalen
Publication date
2020
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English
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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.
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