Skip to main content
SpringerLink
Log in

Dualities and derived equivalences for category \(\mathcal{O}\)

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We determine the Ringel duals for all blocks in the parabolic versions of the BGG category \(\mathcal{O}\) associated to a reductive finite-dimensional Lie algebra. In particular, we find that, contrary to the original category \(\mathcal{O}\) and the specific previously known cases in the parabolic setting, the blocks are not necessarily Ringel self-dual. However, the parabolic category \(\mathcal{O}\) as a whole is still Ringel self-dual. Furthermore, we use generalisations of the Ringel duality functor to obtain large classes of derived equivalences between blocks in parabolic and original category \(\mathcal{O}\). We subsequently classify all derived equivalence classes of blocks of category \(\mathcal{O}\) in type A which preserve the Koszul grading.

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 and E. Lukács, Quasi-hereditary extension algebras, Algebr. Represent. Theory 6 (2003), 97–117.

    Article  MathSciNet  MATH  Google Scholar 

  2. H. H. Andersen and C. Stroppel, Twisting functors on O, Represent. Theory 7 (2003), 681–699 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  3. E. Backelin, Koszul duality for parabolic and singular category O, Represent. Theory 3 (1999), 139–152 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Backelin, The Hom-spaces between projective functors, Represent. Theory 5 (2001), 267–283 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  5. J. N. Bernstein and S. I. Gel’fand, Tensor products of finite- and infinitedimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), 245–285.

    MathSciNet  Google Scholar 

  6. I. N. Bernšteĭn, I. M. Gel’fand and S. I. Gel’fand, A certain category of g-modules, Funkcional. Anal. i Priložen. 10 (1976), 1–8.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Brundan, Symmetric functions, parabolic category O, and the Springer fiber, Duke Math. J. 143 (2008), 41–79.

    Article  MathSciNet  MATH  Google Scholar 

  9. K. J. Carlin, Extensions of Verma modules, Trans. Amer. Math. Soc. 294 (1986), 29–43.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Chen, Poincaré polynomials of hyperquot schemes, Math. Ann. 321 (2001), 235–251.

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Coulembier and V. Mazorchuk, Twisting functors, primitive ideals and star actions for classical Lie superalgebras, J. Reine Ang. Math. (2014).

    Google Scholar 

  12. K. Coulembier and V. Mazorchuk, Some homological properties of category O. IV, ArXiv e-prints (2015).

    MATH  Google Scholar 

  13. K. Coulembier, Bott–Borel–Weil theory and Bernstein–Gel’fand–Gel’fand reciprocity for lie superalgebras, Transformation Groups (2016), 1–43.

    Google Scholar 

  14. J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and sl2-categorification, Ann. of Math. (2) 167 (2008), 245–298.

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Ginzburg, N. Guay, E. Opdam and R. Rouquier, On the category O for rational Cherednik algebras, Invent. Math. 154 (2003), 617–651.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Happel, Triangulated categories in the representation theory of finitedimensional algebras, London Mathematical Society Lecture Note Series, Vol. 119, Cambridge University Press, Cambridge, 1988.

    Google Scholar 

  17. J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category O, Graduate Studies in Mathematics, Vol. 94, American Mathematical Society, Providence, RI, 2008.

    Google Scholar 

  18. R. S. Irving, Projective modules in the category O S : self-duality, Trans. Amer. Math. Soc. 291 (1985), 701–732.

    MathSciNet  MATH  Google Scholar 

  19. J. C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol. 3, Springer-Verlag, Berlin, 1983.

    Google Scholar 

  20. M. Khovanov, Crossingless matchings and the cohomology of (n, n) Springer varieties, Commun. Contemp. Math. 6 (2004), 561–577.

    Article  MathSciNet  MATH  Google Scholar 

  21. O. Khomenko and V. Mazorchuk, On Arkhipov’s and Enright’s functors, Math. Z. 249 (2005), 357–386.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000), 537–592.

    Article  MathSciNet  MATH  Google Scholar 

  24. V. Mazorchuk, Some homological properties of the category O, Pacific J. Math. 232 (2007), 313–341.

    Article  MathSciNet  MATH  Google Scholar 

  25. V. Mazorchuk, Applications of the category of linear complexes of tilting modules associated with the category O, Algebr. Represent. Theory 12 (2009), 489–512.

    Article  MathSciNet  MATH  Google Scholar 

  26. V. Mazorchuk, Some homological properties of the category O. II, Represent. Theory 14 (2010), 249–263.

    Article  MathSciNet  MATH  Google Scholar 

  27. V. Mazorchuk and S. Ovsienko, A pairing in homology and the category of linear complexes of tilting modules for a quasi-hereditary algebra, J. Math. Kyoto Univ. 45 (2005), 711–741, With an appendix by Catharina Stroppel.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. V. Mazorchuk and C. Stroppel, Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module, Trans. Amer. Math. Soc. 357 (2005), 2939–2973 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  30. V. Mazorchuk and C. Stroppel, On functors associated to a simple root, J.Algebra 314 (2007), 97–128.

    Article  MathSciNet  MATH  Google Scholar 

  31. V. Mazorchuk and C. Stroppel, Projective-injective modules, Serre functors and symmetric algebras, J. Reine Angew. Math. 616 (2008), 131–165.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Rocha-Caridi, Splitting criteria for g-modules induced from a parabolic and the Berňsteĭn-Gel’fand-Gel’fand resolution of a finite-dimensional, irreducible g-module, Trans. Amer. Math. Soc. 262 (1980), 335–366.

    MathSciNet  MATH  Google Scholar 

  34. S. Ryom-Hansen, Koszul duality of translation- and Zuckerman functors, J. Lie Theory 14 (2004), 151–163.

    MathSciNet  MATH  Google Scholar 

  35. J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), 436–456.

    Article  MathSciNet  MATH  Google Scholar 

  36. J. Rickard, Translation functors and equivalences of derived categories for blocks of algebraic groups, in Finite-dimensional algebras and related topics (Ottawa, ON, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 255–264.

    Chapter  Google Scholar 

  37. C. M. Ringel, The category of modules with good filtrations over a quasihereditary algebra has almost split sequences, Math. Z. 208 (1991), 209–223.

    Article  MathSciNet  MATH  Google Scholar 

  38. J. C. d. S. O. Santos, Foncteurs de Zuckerman pour les superalgèbres de Lie, J. Lie Theory 9 (1999), 69–112.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  40. W. Soergel, Charakterformeln für Kipp-Moduln über Kac-Moody-Algebren, Represent. Theory 1 (1997), 115–132 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  41. C. Stroppel, Category O: gradings and translation functors, J. Algebra 268 (2003), 301–326.

    Article  MathSciNet  MATH  Google Scholar 

  42. C. Stroppel, Category O: quivers and endomorphism rings of projectives, Represent. Theory 7 (2003), 322–345 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  43. C. Stroppel, Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compos. Math. 145 (2009), 954–992.

    Article  MathSciNet  MATH  Google Scholar 

  44. B. Webster, Knot invariants and higher representation theory, ArXiv e-prints (2013), accepted in Memoirs of the AMS.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Analysis, Ghent University, Krijgslaan 281, 9000, Gent, Belgium

    Kevin Coulembier

  2. Department of Mathematics, University of Uppsala, Box 480, SE-75106, Uppsala, Sweden

    Volodymyr Mazorchuk

Authors
  1. Kevin Coulembier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Volodymyr Mazorchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Volodymyr Mazorchuk.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Coulembier, K., Mazorchuk, V. Dualities and derived equivalences for category \(\mathcal{O}\) . Isr. J. Math. 219, 661–706 (2017). https://doi.org/10.1007/s11856-017-1494-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-017-1494-y

Search

Navigation