Mathematische Zeitschrift

, Volume 282, Issue 1–2, pp 411–434 | Cite as

Isotypic faithful 2-representations of \({\mathcal {J}}\)-simple fiat 2-categories



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.



A substantial part of the paper was written during a visit of the second author to Uppsala University, whose hospitality is gratefully acknowledged. The visit was supported by EPSRC grant EP/K011782/1 and by the Swedish Research Council. The first author is partially supported by the Swedish Research Council. The second author is partially supported by EPSRC grant EP/K011782/1.


  1. 1.
    Agerholm, T.: Simple \(2\)-representations and classification of categorifications. Ph.D. Thesis, Aarhus University (2011)Google Scholar
  2. 2.
    Bernstein, J., Frenkel, I., Khovanov, M.: A categorification of the Temperley-Lieb algebra and Schur quotients of \(U(\mathfrak{sl}_2)\) via projective and Zuckerman functors. Sel. Math. New Ser. 5(2), 199–241 (1999)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bernstein, J., Gelfand, S.: Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras. Compos. Math. 41(2), 245–285 (1980)MATHMathSciNetGoogle Scholar
  4. 4.
    Chuang, J., Rouquier, R.: Derived equivalences for symmetric groups and \(\mathfrak{sl} _2\)-categorification. Ann. of Math. (2) 167(1), 245–298 (2008)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Elias, B., Williamson, G.: The Hodge theory of Soergel bimodules. Preprint. arXiv:1212.0791. To appear in Ann. of Math
  6. 6.
    Ellis, A., Lauda, A.: An odd categorification of quantum \(\mathfrak{sl} (2)\). Preprint. arXiv:1307.7816
  7. 7.
    Etingof, P., Gelaki, S., Nikshych, D., Ostrik, A.V.: Tensor categories. Manuscript, available from
  8. 8.
    Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories. Ann. of Math. (2) 162(2), 581–642 (2005)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Freyd, P.: Representations in abelian categories. In: Proceedings of the Conference Categorical Algebra, pp. 95-120 (1966)Google Scholar
  10. 10.
    Ganter, N., Kapranov, M.: Symmetric and exterior powers of categories. Transform. Groups 19(1), 57–103 (2014)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Kapranov, M., Voevodsky, V.: \(2\)-categories and Zamolodchikov tetrahedra equations. Algebraic groups and their generalizations: quantum and infinite-dimensional methods (University Park, PA, 1991). Proc. Sympos. Pure Math. 56, Part 2, 177-259. American Mathmatical Society, Providence, RI (1994)Google Scholar
  12. 12.
    Khovanov, M., Lauda, A.: A categorification of a quantum \(\mathfrak{sl}_n\). Quantum Topol. 1, 1–92 (2010)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Leinster, T.: Basic bicategories. Preprint. arXiv:math/9810017
  14. 14.
    Losev, I., Webster, B.: On uniqueness of tensor products of irreducible categorifications. Preprint. arXiv:1303.1336
  15. 15.
    Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer, New York (1998)MATHGoogle Scholar
  16. 16.
    Mackaay, M., Thiel, A.-L.: Categorifications of the extended affine Hecke algebra and the affine \(q\). Preprint. arXiv:1302.3102
  17. 17.
    Mazorchuk, V., Miemietz, V.: Cell \(2\)-representations of finitary \(2\)-categories. Compos. Math. 147, 1519–1545 (2011)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Mazorchuk, V., Miemietz, V.: Additive versus abelian \(2\)-representations of fiat \(2\)-categories. Mosc. Math. J. 14(3), 595–615 (2014)MATHMathSciNetGoogle Scholar
  19. 19.
    Mazorchuk, V., Miemietz, V.: Endomorphisms of cell \(2\)-representations. Preprint. arXiv:1207.6236
  20. 20.
    Mazorchuk, V., Miemietz, V.: Morita theory for finitary \(2\)-categories. Preprint. arXiv:1304.4698. To appear in Quantum Topol
  21. 21.
    Mazorchuk, V., Miemietz, V.: Transitive \(2\)-categories. Preprint. arXiv:1404.7589
  22. 22.
    Rouquier, R.: \(2\)-Kac-Moody algebras. Preprint. arXiv:0812.5023
  23. 23.
    Rouquier, R.: Quiver Hecke algebras and \(2\)-Lie algebras. Algebra Colloq. 19, 359–410 (2012)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Sartori, A., Stroppel, C.: Categorification of tensor product representations of \(\mathfrak{sl} (k)\). Preprint. arXiv:1407.4267
  25. 25.
    Soergel, W.: The combinatorics of Harish-Chandra bimodules. J. Reine Angew. Math. 429, 49–74 (1992)MATHMathSciNetGoogle Scholar
  26. 26.
    Xantcha, Q.: Gabriel \(2\)-Categories. Preprint. arXiv:1310.1586, to appear in J. Lond. Math. Soc

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.School of MathematicsUniversity of East AngliaNorwichUK

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