Israel Journal of Mathematics

, Volume 219, Issue 2, pp 661–706 | Cite as

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

  • Kevin Coulembier
  • Volodymyr Mazorchuk


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.


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  1. [ÁDL03]
    I. Ágoston, V. Dlab and E. Lukács, Quasi-hereditary extension algebras, Algebr. Represent. Theory 6 (2003), 97–117.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [AS03]
    H. H. Andersen and C. Stroppel, Twisting functors on O, Represent. Theory 7 (2003), 681–699 (electronic).MathSciNetCrossRefzbMATHGoogle Scholar
  3. [Bac99]
    E. Backelin, Koszul duality for parabolic and singular category O, Represent. Theory 3 (1999), 139–152 (electronic).MathSciNetCrossRefzbMATHGoogle Scholar
  4. [Bac01]
    E. Backelin, The Hom-spaces between projective functors, Represent. Theory 5 (2001), 267–283 (electronic).MathSciNetCrossRefzbMATHGoogle Scholar
  5. [BG80]
    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.MathSciNetGoogle Scholar
  6. [BGG76]
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [BGS96]
    A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473–527.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [Bru08]
    J. Brundan, Symmetric functions, parabolic category O, and the Springer fiber, Duke Math. J. 143 (2008), 41–79.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Car86]
    K. J. Carlin, Extensions of Verma modules, Trans. Amer. Math. Soc. 294 (1986), 29–43.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Che01]
    L. Chen, Poincaré polynomials of hyperquot schemes, Math. Ann. 321 (2001), 235–251.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [CM14]
    K. Coulembier and V. Mazorchuk, Twisting functors, primitive ideals and star actions for classical Lie superalgebras, J. Reine Ang. Math. (2014).Google Scholar
  12. [CM15]
    K. Coulembier and V. Mazorchuk, Some homological properties of category O. IV, ArXiv e-prints (2015).zbMATHGoogle Scholar
  13. [Cou16]
    K. Coulembier, Bott–Borel–Weil theory and Bernstein–Gel’fand–Gel’fand reciprocity for lie superalgebras, Transformation Groups (2016), 1–43.Google Scholar
  14. [CR08]
    J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and sl2-categorification, Ann. of Math. (2) 167 (2008), 245–298.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [GGOR03]
    V. Ginzburg, N. Guay, E. Opdam and R. Rouquier, On the category O for rational Cherednik algebras, Invent. Math. 154 (2003), 617–651.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [Hap88]
    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. [Hum08]
    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. [Irv85]
    R. S. Irving, Projective modules in the category O S: self-duality, Trans. Amer. Math. Soc. 291 (1985), 701–732.MathSciNetzbMATHGoogle Scholar
  19. [Jan83]
    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. [Kho04]
    M. Khovanov, Crossingless matchings and the cohomology of (n, n) Springer varieties, Commun. Contemp. Math. 6 (2004), 561–577.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [KM05]
    O. Khomenko and V. Mazorchuk, On Arkhipov’s and Enright’s functors, Math. Z. 249 (2005), 357–386.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [Lep77]
    J. Lepowsky, A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra 49 (1977), 496–511.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Mat00]
    O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble) 50 (2000), 537–592.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [Maz07]
    V. Mazorchuk, Some homological properties of the category O, Pacific J. Math. 232 (2007), 313–341.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [Maz09]
    V. Mazorchuk, Applications of the category of linear complexes of tilting modules associated with the category O, Algebr. Represent. Theory 12 (2009), 489–512.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [Maz10]
    V. Mazorchuk, Some homological properties of the category O. II, Represent. Theory 14 (2010), 249–263.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [MO05]
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [MOS09]
    V. Mazorchuk, S. Ovsienko and C. Stroppel, Quadratic duals, Koszul dual functors, and applications, Trans. Amer. Math. Soc. 361 (2009), 1129–1172.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [MS05]
    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).MathSciNetCrossRefzbMATHGoogle Scholar
  30. [MS07]
    V. Mazorchuk and C. Stroppel, On functors associated to a simple root, J.Algebra 314 (2007), 97–128.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [MS08]
    V. Mazorchuk and C. Stroppel, Projective-injective modules, Serre functors and symmetric algebras, J. Reine Angew. Math. 616 (2008), 131–165.zbMATHGoogle Scholar
  32. [MVS04]
    R. Martínez Villa and M. Saorín, Koszul equivalences and dualities, Pacific J. Math. 214 (2004), 359–378.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [RC80]
    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.MathSciNetzbMATHGoogle Scholar
  34. [RH04]
    S. Ryom-Hansen, Koszul duality of translation- and Zuckerman functors, J. Lie Theory 14 (2004), 151–163.MathSciNetzbMATHGoogle Scholar
  35. [Ric89]
    J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), 436–456.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [Ric94]
    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.CrossRefGoogle Scholar
  37. [Rin91]
    C. M. Ringel, The category of modules with good filtrations over a quasihereditary algebra has almost split sequences, Math. Z. 208 (1991), 209–223.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [San99]
    J. C. d. S. O. Santos, Foncteurs de Zuckerman pour les superalgèbres de Lie, J. Lie Theory 9 (1999), 69–112.MathSciNetzbMATHGoogle Scholar
  39. [Soe90]
    W. Soergel, Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421–445.MathSciNetzbMATHGoogle Scholar
  40. [Soe97]
    W. Soergel, Charakterformeln für Kipp-Moduln über Kac-Moody-Algebren, Represent. Theory 1 (1997), 115–132 (electronic).MathSciNetCrossRefzbMATHGoogle Scholar
  41. [Str03a]
    C. Stroppel, Category O: gradings and translation functors, J. Algebra 268 (2003), 301–326.MathSciNetCrossRefzbMATHGoogle Scholar
  42. [Str03b]
    C. Stroppel, Category O: quivers and endomorphism rings of projectives, Represent. Theory 7 (2003), 322–345 (electronic).MathSciNetCrossRefzbMATHGoogle Scholar
  43. [Str09]
    C. Stroppel, Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compos. Math. 145 (2009), 954–992.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [Web13]
    B. Webster, Knot invariants and higher representation theory, ArXiv e-prints (2013), accepted in Memoirs of the AMS.Google Scholar

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© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisGhent UniversityGentBelgium
  2. 2.Department of MathematicsUniversity of UppsalaUppsalaSweden

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