Skip to main content
Log in

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

  • Published:
Ukrainian Mathematical Journal Aims and scope

It is shown that any simple transitive 2-representation of the 2-category of projective endofunctors for the quiver algebra of đť•‚ and for the quiver algebra of đť•‚ is equivalent to a cell 2-representation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. T. Agerholm, Simple 2-Representations and Classification of Categorifications: PhD Thesis, Ă…rhus University, Denmark (2011).

  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).

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Chuang and R. Rouquier, “Derived equivalences for symmetric groups and sl2-categorification,” Ann. Math., 167, No. 1, 245–298 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Flor, “On groups of non-negative matrices,” Compos. Math., 21, 376–382 (1969).

    MathSciNet  MATH  Google Scholar 

  5. A. L. Grensing and V. Mazorchuk, “Categorification of the Catalan monoid,” Semigroup Forum, 89, No. 1, 155–168 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. L. Grensing and V. Mazorchuk, “Finitary 2-categories associated with dual projection functors,” Comm. Contemp. Math., 19, No. 3, 40 p. (2017).

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Kildetoft and V. Mazorchuk, “Parabolic projective functors in type A,” Adv. Math., 301, 785–803 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Kudryavtseva and V. Mazorchuk, “On multisemigroups,” Port. Math., 72, No. 1, 47–80 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  9. T. Leinster, Basic Bicategories, Preprint arXiv:math/9810017.

  10. S. Mac Lane, Categories for the Working Mathematician, Springer Verlag (1998).

  11. V. Mazorchuk, Lectures on Algebraic Categorification, in: European Mathematical Society, QGM Master Class Series (2012), 128 p.

  12. V. Mazorchuk and V. Miemietz, “Cell 2-representations of finitary 2-categories,” Compos. Math., 147, 1519–1545 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  13. V. Mazorchuk and V. Miemietz, “Additive versus abelian 2-representations of fiat 2-categories,” Mosc. Math. J., 14, No. 3, 595–615 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. Mazorchuk and V. Miemietz, “Endmorphisms of cell 2-representations,” Int. Math. Res. Not. IMRN, 2016, No. 24, 7471–7498 (2016).

    Article  MATH  Google Scholar 

  15. V. Mazorchuk and V. Miemietz, “Morita theory for finitary 2-categories,” Quantum Topol., 7, No. 1, 1–28 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. Mazorchuk and V. Miemietz, “Transitive 2-representations of finitary 2-categories,” Trans. Amer. Math. Soc., 368, No. 11, 7623–7644 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Rouquier, Categorification of the Braid Groups, Preprint arXiv:math/0409593.

  19. R. Rouquier, 2-Kac–Moody Algebras, Preprint arXiv:0812.5023.

  20. Q. Xantcha, “Gabriel 2-quivers for finitary 2-categories,” J. Lond. Math. Soc. (2), 92, No. 3, 615–632 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  21. X. Zhang, “Duflo involutions for 2-categories associated to tree quivers,” J. Algebra Appl., 15, No. 3, 25 p. (2016).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Zimmermann, “Simple transitive 2-representations of Soergel bimodules in type B2,” J. Pure Appl. Algebra, 221, No. 3, 666–690 (2017).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 12, pp. 1625–1649, December, 2018.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mazorchuk, V., Zhang, X. Simple Transitive 2-Representations for Two Nonfiat 2-Categories of Projective Functors. Ukr Math J 70, 1873–1900 (2019). https://doi.org/10.1007/s11253-019-01615-w

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-019-01615-w

Navigation