Skip to main content
SpringerLink
Log in

Recollements of self-injective algebras, and classification of self-injective diagram algebras

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Diagram algebras, in particular Brauer algebras, Birman–Murakami-Wenzl algebras and partition algebras, are used in representation theory and invariant theory of orthogonal and symplectic groups, in knot theory, in mathematical physics and elsewhere. Classifications are known when such algebras are semisimple, of finite global dimension or quasi-hereditary. We obtain a characterisation of the self-injective case, which is shown to coincide with the (previously also unknown) symmetric case. The main tool is to show that indecomposable self-injective algebras in general are derived simple, that is, their bounded derived module categories admit trivial recollements only. As a consequence, self-injective algebras are seen to satisfy a derived Jordan–Hölder theorem.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abe, H., Hoshino, M.: Derived equivalences and Gorenstein algebras. J. Pure Appl. Algebra 211(1), 55–69 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Al-Nofayee, S., Rickard, J.: Rigidity of tilting complexes and derived equivalence for self-injective algebras. arXiv:1311.0504

  3. Angeleri Hügel, L., Koenig, S., Liu, Q., Yang, D.: Stratifying derived module categories. C.R. Acad. Sci. Paris Sér. I Math. 349(21–22), 1139–1144

  4. Angeleri Hügel, L., Koenig, S., Liu, Q.: Recollements and tilting objects. J. Pure. Appl. Algebra 215, 420–438 (2011)

  5. Angeleri Hügel, L., Koenig, S., Liu, Q.: On the uniqueness of stratifications of derived module categories. J. Algebra 359, 120–137 (2012)

  6. Angeleri Hügel, L., Koenig, S., Liu, Q.: Jordan-Hölder theorems for derived module categories of piecewise hereditary algebras. J. Algebra 352, 361–381 (2012)

  7. Angeleri Hügel, L., Koenig, S., Liu, Q., Yang, D.: Ladders and simplicity of derived module categories. J. Algebra 472, 15–66 (2017)

  8. Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astér., vol. 100, Soc. Math. France, (1982)

  9. Chen, X.W.: Singularity categories, Schur functors and triangular matrix rings. Algebras Represent. Theory 12, 181–191 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Graham, J., Lehrer, G.: Cellular algebras. Invent. Math. 123, 1–34 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Halverson, T., Ram, A.: Partition algebras. Eur. J. Comb. 26(6), 869–921 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hartmann, R., Henke, A., Koenig, S., Paget, R.: Cohomological stratification of diagram algebras. Math. Ann. 347, 765–804 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kalck, M.: Derived categories of quasi-hereditary algebras and their derived composition series. In: Representation Theory—Current Trends and Perspectives, pp. 269–308. European Mathematical Society Publishing House (2017). doi:10.4171/171-1/11

  14. Koenig, S.: A panorama of diagram algebras. Trends in representation theory of algebras and related topics, 491–540, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, (2008)

  15. Koenig, S., Xi, C.C.: When is a cellular algebra quasi-hereditary? Math. Ann. 315, 281–293 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Koenig, S., Xi, C.C.: A self-injective cellular algebra is weakly symmetric. J. Algebra 228(1), 51–59 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Koenig, S., Xi, C.C.: A characteristic free approach to Brauer algebras. Trans. Am. Math. Soc. 353, 1489–1505 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Liu, Q., Yang, D.: Blocks of group algebras are derived simple. Math. Z. 272, 913–920 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Pan, S.: Recollements and Gorenstein algebras. Int. J. Algebra 7(17–20), 829–832 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  20. Qin, Y., Han, Y.: Reducing homological conjectures by n-recollements. Algebras Represent. Theory 19, 377–395 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. 39(3), 436–456 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rui, H.: A criterion on the semisimple Brauer algebras. J. Comb. Theory Ser. A 111(1), 78–88 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Rui, H., Si, M.: A criterion on the semisimple Brauer algebras. II. J. Comb. Theory Ser. A 113(6), 1199–1203 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Rui, H., Si, M.: Gram determinants and semisimplicity criteria for Birman–Wenzl algebras. J. Reine Angew. Math. 631, 153–179 (2009)

    MATH  MathSciNet  Google Scholar 

  25. Rui, H., Si, M.: Singular parameters for the Birman–Murakami–Wenzl algebra. J. Pure Appl. Algebra 216(6), 1295–1305 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wenzl, H.: On the structure of Brauer’s centralizer algebras. Ann. of Math. 128(2), 173–193 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  27. Wiedemann, A.: On stratifications of derived module categories. Can. Math. Bull. 34(2), 275–280 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  28. Xi, C.C.: Partition algebras are cellular. Compositio Math. 119(1), 99–109 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  29. Xi, C.C.: On the quasi-heredity of Birman–Wenzl algebras. Adv. Math. 154(2), 280–298 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

This paper was written during the first author’s visit to Stuttgart in 2016. She would like to express her gratitude to the second author for hospitality and many useful discussions. The first author’s research was supported by Grants from CSC, NSFC No. 11301398 and RFDP No. 20130141120035.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Yiping Chen

  2. Institute of Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Steffen Koenig

Authors
  1. Yiping Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Steffen Koenig
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Steffen Koenig.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Y., Koenig, S. Recollements of self-injective algebras, and classification of self-injective diagram algebras. Math. Z. 287, 1009–1027 (2017). https://doi.org/10.1007/s00209-017-1857-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-017-1857-4

Keywords

Mathematics Subject Classification

Search

Navigation