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Simple Transitive 2-Representations for Two Nonfiat 2-Categories of Projective Functors

  • V. Mazorchuk
  • X. Zhang
Article

It is shown that any simple transitive 2-representation of the 2-category of projective endofunctors for the quiver algebra of 𝕂 Open image in new window and for the quiver algebra of 𝕂 Open image in new window is equivalent to a cell 2-representation.

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References

  1. 1.
    T. Agerholm, Simple 2-Representations and Classification of Categorifications: PhD Thesis, Århus University, Denmark (2011).Google Scholar
  2. 2.
    J. Bernstein, I. Frenkel, and M. Khovanov, “A categorification of the Temperley–Lieb algebra and Schur quotients of U(sl2) via projective and Zuckerman functors,” Selecta Math. (N. S.), 5, No. 2, 199–241 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. Chuang and R. Rouquier, “Derived equivalences for symmetric groups and sl2-categorification,” Ann. Math., 167, No. 1, 245–298 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    P. Flor, “On groups of non-negative matrices,” Compos. Math., 21, 376–382 (1969).MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. L. Grensing and V. Mazorchuk, “Categorification of the Catalan monoid,” Semigroup Forum, 89, No. 1, 155–168 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. L. Grensing and V. Mazorchuk, “Finitary 2-categories associated with dual projection functors,” Comm. Contemp. Math., 19, No. 3, 40 p. (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    T. Kildetoft and V. Mazorchuk, “Parabolic projective functors in type A,Adv. Math., 301, 785–803 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    G. Kudryavtseva and V. Mazorchuk, “On multisemigroups,” Port. Math., 72, No. 1, 47–80 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    T. Leinster, Basic Bicategories, Preprint arXiv:math/9810017.Google Scholar
  10. 10.
    S. Mac Lane, Categories for the Working Mathematician, Springer Verlag (1998).Google Scholar
  11. 11.
    V. Mazorchuk, Lectures on Algebraic Categorification, in: European Mathematical Society, QGM Master Class Series (2012), 128 p.Google Scholar
  12. 12.
    V. Mazorchuk and V. Miemietz, “Cell 2-representations of finitary 2-categories,” Compos. Math., 147, 1519–1545 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    V. Mazorchuk and V. Miemietz, “Additive versus abelian 2-representations of fiat 2-categories,” Mosc. Math. J., 14, No. 3, 595–615 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    V. Mazorchuk and V. Miemietz, “Endmorphisms of cell 2-representations,” Int. Math. Res. Not. IMRN, 2016, No. 24, 7471–7498 (2016).CrossRefzbMATHGoogle Scholar
  15. 15.
    V. Mazorchuk and V. Miemietz, “Morita theory for finitary 2-categories,” Quantum Topol., 7, No. 1, 1–28 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    V. Mazorchuk and V. Miemietz, “Transitive 2-representations of finitary 2-categories,” Trans. Amer. Math. Soc., 368, No. 11, 7623–7644 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    V. Mazorchuk and V. Miemietz, “Isotypic faithful 2-representations of J -simple fiat 2-categories,” Math. Z., 282, No. 1-2, 411–434 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    R. Rouquier, Categorification of the Braid Groups, Preprint arXiv:math/0409593.Google Scholar
  19. 19.
    R. Rouquier, 2-Kac–Moody Algebras, Preprint arXiv:0812.5023.Google Scholar
  20. 20.
    Q. Xantcha, “Gabriel 2-quivers for finitary 2-categories,” J. Lond. Math. Soc. (2), 92, No. 3, 615–632 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    X. Zhang, “Duflo involutions for 2-categories associated to tree quivers,” J. Algebra Appl., 15, No. 3, 25 p. (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    X. Zhang, “Simple transitive 2-representations and Drinfeld center for some finitary 2-categories,” J. Pure Appl. Algebra, 222, No. 1, 97–130 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    J. Zimmermann, “Simple transitive 2-representations of Soergel bimodules in type B2,” J. Pure Appl. Algebra, 221, No. 3, 666–690 (2017).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • V. Mazorchuk
    • 1
  • X. Zhang
    • 2
  1. 1.Uppsala UniversityUppsalaSweden
  2. 2.East China Normal UniversityShanghaiChina

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