Abstract:
We study connections between additive and abelian 22-representations of fiat 22-categories, describe combinatorics of 22-categories in terms of multisemigroups and determine the annihilator of a cell 22-representation. We also describe, in detail, examples of fiat 22-categories associated to sl2-categorification in the sense of Chuang and Rouquier, and 2-categorical analogues of Schur algebras.
Key words and phrases:2-category, 2-representation, functor, natural transformation, additive category, abelian category.
Received:January 10, 2012; in revised form February 26, 2014
\Bibitem{MazMie14}
\by Volodymyr~Mazorchuk, Vanessa~Miemietz
\paper Additive versus abelian $2$-representations of fiat $2$-categories
\jour Mosc. Math.~J.
\yr 2014
\vol 14
\issue 3
\pages 595--615
\mathnet{http://mi.mathnet.ru/mmj534}
\crossref{https://doi.org/10.17323/1609-4514-2014-14-3-595-615}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3241761}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000342789400007}
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Helena Jonsson, “J-equivalence for associative algebras”, Beitr Algebra Geom, 2024
Helena Jonsson, “On Simple Transitive 2-representations of Bimodules over the Dual Numbers”, Algebr Represent Theor, 26:5 (2023), 2057
Hankyung Ko, Volodymyr Mazorchuk, “2-representations of small quotients of Soergel bimodules in infinite types”, Proc. Amer. Math. Soc., 2023
Mateusz Stroiński, “Weighted Colimits of 2-Representations and Star Algebras”, Appl Categor Struct, 31:4 (2023)
Katerina Hristova, Vanessa Miemietz, “Basic Hopf algebras and symmetric bimodules”, Journal of Pure and Applied Algebra, 227:7 (2023), 107328
Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer, Xiaoting Zhang, “Simple transitive 2‐representations of Soergel bimodules for finite Coxeter types”, Proceedings of London Math Soc, 126:5 (2023), 1585
Ko H., Mazorchuk V., Zhang X., “Adjunction in the Absence of Identity”, Appl. Categ. Struct., 30:1 (2022), 123–172
Mateusz Stroiński, “Simple transitive 2-representations of 2-categories associated to self-injective cores”, Communications in Algebra, 50:8 (2022), 3630
Chan A., Miemietz V., “Coidempotent Subcoalgebras and Short Exact Sequences of Finitary 2-Representations”, Nagoya Math. J., 243 (2021), PII S002776302000001X, 316–341
Kevin Coulembier, Michael Ehrig, “The Periplectic Brauer Algebra III: The Deligne Category”, Algebr Represent Theor, 24:4 (2021), 993
Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer, Xiaoting Zhang, “Finitary birepresentations of finitary bicategories”, Forum Mathematicum, 33:5 (2021), 1261
Jonsson H., “Cell Structure of Bimodules Over Radical Square Zero Nakayama Algebras”, Commun. Algebr., 48:2 (2020), 573–588
Volodymyr Mazorchuk, Xiaoting Zhang, “Bimodules over uniformly oriented A n quivers with radical square zero”, Kyoto J. Math., 60:3 (2020)
Volodymyr Mazorchuk, Vanessa Miemietz, Xiaoting Zhang, “2-categories of symmetric bimodules and their
2-representations”, Pacific J. Math., 306:2 (2020), 645
Robert Laugwitz, Vanessa Miemietz, “Cell 2-representations and categorification at prime roots of unity”, Advances in Mathematics, 361 (2020), 106937
Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz, Xiaoting Zhang, “ANALOGUES OF CENTRALIZER SUBALGEBRAS FOR FIAT 2-CATEGORIES AND THEIR 2-REPRESENTATIONS”, J. Inst. Math. Jussieu, 19:6 (2020), 1793
V. Mazorchuk, X. Zhang, “Simple transitive 2-representations for two nonfiat 2-categories of projective functors”, Ukr. Math. J., 70:12 (2019), 1873–1900
T. Kildetoft, M. Mackaay, V. Mazorchuk, J. Zimmermann, “Simple transitive 2-representations of small quotients of Soergel bimodules”, Trans. Am. Math. Soc., 371:8 (2019), 5551–5590
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