Skip to main content
SpringerLink
Log in

Highest weight categories arising from Khovanov's diagram algebra II: Koszulity

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

This is the second of a series of four papers studying various generalisations of Khovanov's diagram algebra. In this paper we develop the general theory of Khovanov's diagrammatically defined “projective functors” in our setting. As an application, we give a direct proof of the fact that the quasi-hereditary covers of generalised Khovanov algebras are Koszul.

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. I. Ágoston, V. Dlab, E. Lukács, Quasi-hereditary extension algebras, Algebr. Represent. Theory 6 (2003), 97–117.

    Article  MathSciNet  MATH  Google Scholar 

  2. E. Backelin, Koszul duality for parabolic and singular category \( \mathcal{O} \), Represent. Theory 3 (1999), 139–152.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Beilinson, J. Bernstein, Localisation de \( \mathfrak{g} \) -modules, C. R. Acad. Sci. Paris Ser. I Math. 292 (1981), 15–18.

    MathSciNet  MATH  Google Scholar 

  4. J. Bernstein, S. Gelfand, Tensor products of finite and infinite representations of semisimple Lie algebras, Compositio Math. 41 (1980), 245–285.

    MathSciNet  MATH  Google Scholar 

  5. A. Beilinson, V. Ginzburg, W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473–527.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Bernstein, I. Gelfand, S. Gelfand, Differential operators on the base affine space and a study of \( \mathfrak{g} \)-modules, in: Lie Groups and their Representations (Budapest, 1971), Halsted, New York, 1975, pp. 21–64.

  7. B. Boe, M. Hunziker, Kostant modules in blocks of category \( {\mathcal{O}_{\text{S}}} \), Comm. Alg. 37 (2009), 323–356.

    Article  MathSciNet  MATH  Google Scholar 

  8. T. Braden, Perverse sheaves on Grassmannians, Canad. J. Math. 54 (2002), 493–532.

    MathSciNet  MATH  Google Scholar 

  9. J. Brundan, Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra \( \mathfrak{g}\mathfrak{l}\left( {m\left| n \right.} \right) \), J. Amer. Math. Soc. 16 (2003), 185–231.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Brundan, C. Stroppel, Highest weight categories arising from Khovanov's diagram algebra I: Cellularity; arXiv:0806.1532.

  11. S.-J. Cheng, J.-H. Kwon, N. Lam, A BGG-type resolution for tensor modules over general linear superalgebra, Lett. Math. Phys. 84 (2008), 75–87.

    Article  MathSciNet  MATH  Google Scholar 

  12. E. Cline, B. Parshall, L. Scott, The homological dual of a highest weight category, Proc. London Math. Soc. 68 (1994), 296–316.

    MathSciNet  Google Scholar 

  13. I. Heckenberger, S. Kolb, On the Bernstein–Gelfand–Gelfand resolution for Kac-Moody algebras and quantised enveloping algebras, Transform. Groups 12 (2007), 647–655.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. Irving, Projective modules in the category \( {\mathcal{O}_{\text{S}}} \): Self-duality, Trans. Amer. Math. Soc. 291 (1985), 701–732.

    Article  MathSciNet  Google Scholar 

  15. D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Khovanov, A functor-valued invariant of tangles, Alg. Geom. Topology 2 (2002), 665–741.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Koenig, H. Slungard, C. Xi, Double centraliser properties, dominant dimension and tilting modules, J. Algebra 240 (2001), 393–412.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Lascoux, M.-P. Schützenberger, Polynômes de Kazhdan et Lusztig pour les Grassmanniennes, Astérisque 87–88 (1981), 249–266.

    Google Scholar 

  19. J. Lepowsky, A generalization of the Bernstein–Gelfand–Gelfand resolution, J. Algebra 49 (1977), 496–511.

    Article  MathSciNet  MATH  Google Scholar 

  20. V. Mazorchuk, S. Ovsienko, C. Stroppel, Quadratic duals, Koszul dual functors, and applications, Trans. Amer. Math. Soc. 361 (2009), 1129–1172

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Martínez Villa, M. Saorín, Koszul equivalences and dualities, Pacific J. Math. 214 (2004), 359–378.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. Rouquier, q-Schur algebras and complex reection groups, Mosc. Math. J. 8 (2008), 119–158.

    MathSciNet  MATH  Google Scholar 

  23. W. Soergel, Kategorie \( \mathcal{O} \) , perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421–445.

    Article  MathSciNet  MATH  Google Scholar 

  24. C. Stroppel, Category \( \mathcal{O} \): Quivers and endomorphism rings of projectives, Represent. Theory 7 (2003), 322–345.

    Article  MathSciNet  MATH  Google Scholar 

  25. C. Stroppel, Categorification of the Temperley–Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126 (2005), 547–596.

    Article  MathSciNet  MATH  Google Scholar 

  26. C. Stroppel, TQFT with corners and tilting functors in the Kac–Moody case, arXiv:math/0605103.

  27. C. Stroppel, Perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compositio Math. 145 (2009), 954–992.

    Article  MathSciNet  MATH  Google Scholar 

  28. D. Vogan, Irreducible representations of semisimple Lie groups II: The Kazhdan–Lusztig conjectures, Duke Math. J. 46 (1979), 805–859.

    Article  MathSciNet  MATH  Google Scholar 

  29. C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, Cambridge, 1994.

    MATH  Google Scholar 

  30. A. B. Зелевий, Мапые разршенця особенносmеú мноƨообразuú IIIyберma,Φyнкц. анализ и его прилож. 17 (1983), no. 2, 75–77. Engl. transl.: A. Zelevinskij, Small resolutions of singularities of Schubert varieties, Funct. Anal. Appl. 17 (1983), 142–144.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Jonathan Brundan

  2. Department of Mathematics, University of Bonn, 53115, Bonn, Germany

    Catharina Stroppel

Authors
  1. Jonathan Brundan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Catharina Stroppel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonathan Brundan.

Additional information

Supported in part by NSF grant no. DMS-0654147.

Supported by the NSF and the Minerva Research Foundation DMS-0635607.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brundan, J., Stroppel, C. Highest weight categories arising from Khovanov's diagram algebra II: Koszulity. Transformation Groups 15, 1–45 (2010). https://doi.org/10.1007/s00031-010-9079-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-010-9079-4

Keywords and phrases

AMS classification

Search

Navigation