460 research outputs found

    Cell 2-representations of finitary 2-categories

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    We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2-representations inspired by Kazhdan-Lusztig cell modules for Hecke algebras. Under some natural assumptions we show that cell 2-representations are strongly simple and do not depend on the choice of a right cell inside a two-sided cell. This reproves and extends the uniqueness result on categorification of Kazhdan-Lusztig cell modules for Hecke algebras of type AA from \cite{MS2}.Comment: 25 page

    Transitive 2-representations of finitary 2-categories

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    In this article, we define and study the class of simple transitive 22-representations of finitary 22-categories. We prove a weak version of the classical Jordan-H{\"o}lder Theorem where the weak composition subquotients are given by simple transitive 22-representations. For a large class of finitary 22-categories we prove that simple transitive 22-representations are exhausted by cell 22-representations. Finally, we show that this large class contains finitary quotients of 22-Kac-Moody algebras

    Isotypic faithful 2-representations of J-simple fiat 2-categories

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    We introduce the class of isotypic 2-representations for finitary 2-categories and the notion of inflation of 2-representations. Under some natural assumptions we show that isotypic 2-representations are equivalent to inflations of cell 2-representations

    2-representations of small quotients of Soergel bimodules in infinite types

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    We determine for which Coxeter types the associated small quotient of the 22-category of Soergel bimodules is finitary and, for such a small quotient, classify the simple transitive 22-representations (sometimes under the additional assumption of gradability). We also describe the underlying categories of the simple transitive 22-representations. For the small quotients of general Coxeter types, we give a description for the cell 22-representations

    Cell 2-representations and categorification at prime roots of unity

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    Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2- categories, enriched with a p-differential, which satisfy finiteness conditions analogous to those of finitary or fiat 2-categories. We construct cell 2-representations in this setup, and consider a class of 2-categories stemming from bimodules over a p-dg category in detail. This class is of particular importance in the categorification of quantum groups, which allows us to apply our results to cyclotomic quotients of the categorifications of small quantum group of type sl2 at prime roots of unity by Elias–Qi [Advances in Mathematics 288 (2016)]. Passing to stable 2-representations gives a way to construct triangulated 2-representations, but our main focus is on working with p-dg enriched 2-representations that should be seen as a p-dg enhancement of these triangulated ones
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