Mathematische Zeitschrift

, Volume 249, Issue 2, pp 357–386 | Cite as

On Arkhipov’s and Enright’s functors

  • Oleksandr Khomenko
  • Volodymyr Mazorchuk


We give a description of Arkhipov’s and (Joseph’s and Deodhar-Mathieu’s versions of) Enright’s endofunctors on the category Open image in new window associated with a fixed triangular decomposition of a complex finite-dimensional semi-simple Lie algebra, in terms of (co)approximation functors with respect to suitably chosen injective (resp. projective) modules. We establish some new connections between these functors, for example we show that Arkhipov’s and Joseph’s functors are adjoint to each other. We also give several proofs of braid relations for Arkhipov’s and Enright’s functors.


Approximation Functor Triangular Decomposition Braid Relation 
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  1. 1.
    Andersen, H.H., Lauritzen, N.: Twisted Verma modules. In: Studies in Memory of Issai Schur, Vol. 210, Progress in Math., Birkhäuser, Basel, 2002, pp. 1–26Google Scholar
  2. 2.
    Andersen, H.H., Stroppel, C.: Twisting functors on Open image in new window Represent. Theory 7, 681–699 (2003)Google Scholar
  3. 3.
    Arkhipov, S.: Semi-infinite cohomology of associative algebras and bar duality. Internat. Math. Res. Notices 17, 833–863 (1997)Google Scholar
  4. 4.
    Arkhipov, S.: Algebraic construction of contragradient quasi-Verma modules in positive characteristics. Preprint MPI 2001-34, Max-Planck Institute für Mathematik, 2001Google Scholar
  5. 5.
    Auslander, M.: Representation theory of Artin algebras I. Commun. Alg. 1, 177–268 (1974)zbMATHGoogle Scholar
  6. 6.
    Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86, 111–152 (1991)zbMATHGoogle Scholar
  7. 7.
    Bernstein, I.N., Gelfand, I.M., Gelfand, S.I.: A certain category of Open image in new window-modules. (Russian) Funkcional. Anal. I Prilozen. 10 (2), 1–8 (1976)Google Scholar
  8. 8.
    Bernstein, J.N., Gelfand, S.I.: Tensor products of finite- and infinite-dimensional representations of semi-simple Lie algebras. Compositio Math. 41, 245–285 (1980)zbMATHGoogle Scholar
  9. 9.
    Bouaziz, A.: Sur les représentations des algèbres de Lie semi-simples construites par T. Enright. In: Non-commutative harmonic analysis and Lie groups. Vol. 880, Springer LN Mathematics, 1981, pp. 57–68Google Scholar
  10. 10.
    Deodhar, V.V.: On a construction of representations and a problem of Enright. Invent. Math. 57, 101–118 (1980)zbMATHGoogle Scholar
  11. 11.
    Enright, T.J.: On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions and multiplicity formulae. Ann. Math. 110, 1–82 (1979)Google Scholar
  12. 12.
    Futorny, V., König, S., Mazorchuk, V.: A combinatorial description of blocks in Open image in new window associated with Open image in new window-induction. J. Algebra 231 (1), 86–103 (2000)Google Scholar
  13. 13.
    Jantzen, J.C.: Einhüllende Algebren halbeinfacher Lie-Algebren. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 3, Springer-Verlag, Berlin-New York, 1983Google Scholar
  14. 14.
    Joseph, A.: The Enright functor on the Bernstein-Gelfand-Gelfand category Open image in new window. Invent. Math. 67 (3), 423–445 (1982)Google Scholar
  15. 15.
    Joseph, A.: Completion functors in the Open image in new window category. Noncommutative harmonic analysis and Lie groups (Marseille, 1982), Lecture Notes in Math., 1020, Springer, Berlin, 1983, pp. 80–106Google Scholar
  16. 16.
    A. Joseph, Quantum groups and their primitive ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 29, Springer-Verlag, Berlin, 1995Google Scholar
  17. 17.
    König, S., Mazorchuk, N.: Enright’s completions and injectively copresented modules. Trans. Am. Math. Soc. 354 (7), 2725–2743 (2002)CrossRefGoogle Scholar
  18. 18.
    Mathieu, O.: Classification of irreducible weight modules. Ann. Inst. Fourier (Grenoble) 50, 537–592 (2000)zbMATHGoogle Scholar
  19. 19.
    Mazorchuk, V., Stroppel, C.: On functors associated to a simple root, Preprint 2004:15, Uppsala University, Uppsala, SwedenGoogle Scholar
  20. 20.
    Rocha-Caridi, A., Wallach, N.R.: Projective modules over graded Lie algebras. I. Math. Z. 180 (2), 151–177 (1982)zbMATHGoogle Scholar
  21. 21.
    Soergel, W.: Charakterformeln für Kipp-Moduln über Kac-Moody-Algebren. Represent. Theory 1, 115–132 (1997)CrossRefzbMATHGoogle Scholar
  22. 22.
    Soergel, W.: Kategorie Open image in new window, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. J. Am. Math. Soc. 3 (2), 421–445 (1990)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Mathematisches Institut der Universität FreiburgFreiburg im BreisgauGermany
  2. 2.Department of MathematicsUppsala UniversityUppsalaSweden

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