Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 016, 17 pages      arXiv:1409.5717
Contribution to the Special Issue on New Directions in Lie Theory

Extension Fullness of the Categories of Gelfand-Zeitlin and Whittaker Modules

Kevin Coulembier a and Volodymyr Mazorchuk b
a) Department of Mathematical Analysis, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
b) Department of Mathematics, Uppsala University, Box 480, SE-751 06, Uppsala, Sweden

Received September 25, 2014, in final form February 20, 2015; Published online February 24, 2015

We prove that the categories of Gelfand-Zeitlin modules of $\mathfrak{g}=\mathfrak{gl}_n$ and Whittaker modules associated with a semi-simple complex finite-dimensional algebra $\mathfrak{g}$ are extension full in the category of all $\mathfrak{g}$-modules. This is used to estimate and in some cases determine the global dimension of blocks of the categories of Gelfand-Zeitlin and Whittaker modules.

Key words: extension fullness; Gelfand-Zeitlin modules; Whittaker modules; Yoneda extensions; homological dimension.

pdf (489 kb)   tex (26 kb)


  1. Baer R., Abelian groups that are direct summands of every containing abelian group, Bull. Amer. Math. Soc. 46 (1940), 800-806.
  2. Bagci I., Christodoulopoulou K., Wiesner E., Whittaker categories and Whittaker modules for Lie superalgebras, Comm. Algebra 42 (2014), 4932-4947, arXiv:1201.5350.
  3. Batra P., Mazorchuk V., Blocks and modules for Whittaker pairs, J. Pure Appl. Algebra 215 (2011), 1552-1568, arXiv:0910.3540.
  4. Benkart G., Ondrus M., Whittaker modules for generalized Weyl algebras, Represent. Theory 13 (2009), 141-164, arXiv:0803.3570.
  5. Brüstle Th., König S., Mazorchuk V., The coinvariant algebra and representation types of blocks of category ${\mathcal O}$, Bull. London Math. Soc. 33 (2001), 669-681.
  6. Christodoulopoulou K., Whittaker modules for Heisenberg algebras and imaginary Whittaker modules for affine Lie algebras, J. Algebra 320 (2008), 2871-2890.
  7. Coulembier K., Mazorchuk V., Some homological properties of the category ${\mathcal O}$. III, arXiv:1404.3401.
  8. Dahlberg R.P., Injective hulls of Lie modules, J. Algebra 87 (1984), 458-471.
  9. Dixmier J., Enveloping algebras, Graduate Studies in Mathematics, Vol. 11, Amer. Math. Soc., Providence, RI, 1996.
  10. Donkin S., On the Hopf algebra dual of an enveloping algebra, Math. Proc. Cambridge Philos. Soc. 91 (1982), 215-224.
  11. Drozd Yu.A., Representations of Lie algebras ${\mathfrak{sl}}(2)$, Visnik Kiïv. Univ. Ser. Mat. Mekh. (1983), no. 25, 70-77.
  12. Drozd Yu.A., Futorny V.M., Ovsienko S.A., Harish-Chandra subalgebras and Gel'fand-Zetlin modules, 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, 79-93.
  13. Drozd Yu.A., Ovsienko S.A., Futorny V.M., Irreducible weighted ${\rm sl}(3)$-modules, Funct. Anal. Appl. 23 (1989), 217-218.
  14. Drozd Yu.A., Ovsienko S.A., Futorny V.M., On Gel'fand-Zetlin modules, Rend. Circ. Mat. Palermo (2) Suppl. (1991), no. 26, 143-147.
  15. Eckmann B., Schopf A., Über injektive Moduln, Arch. Math. 4 (1953), 75-78.
  16. Feldvoss J., Injective modules and prime ideals of universal enveloping algebras, in Abelian Groups, Rings, Modules, and Homological Algebra, Lect. Notes Pure Appl. Math., Vol. 249, Chapman & Hall/CRC, Boca Raton, FL, 2006, 107-119, math.RT/0504539.
  17. Fuser A., Autour de la conjecture d'Alexandru, Ph.D. Thesis, Université de Nancy, Nancy, 1997.
  18. Futorny V., Grantcharov D., Ramirez L.E., Singular Gelfand-Tsetlin modules of $\mathfrak{gl}(n)$, arXiv:1409.0550.
  19. Futorny V., Nakano D.K., Pollack R.D., Representation type of the blocks of category ${\mathcal O}$, Q. J. Math. 52 (2001), 285-305.
  20. Futorny V., Ovsienko S., Kostant's theorem for special filtered algebras, Bull. London Math. Soc. 37 (2005), 187-199, math.RA/0303372.
  21. Futorny V., Ovsienko S., Fibers of characters in Gelfand-Tsetlin categories, Trans. Amer. Math. Soc. 366 (2014), 4173-4208, math.RT/0610071.
  22. Gaillard P.-Y., Introduction to the Alexandru conjecture, math.RT/0003069.
  23. Gel'fand I.M., Tsetlin M.L., Finite-dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk USSR 71 (1950), 825-828.
  24. Guo X., Liu X., Whittaker modules over generalized Virasoro algebras, Comm. Algebra 39 (2011), 3222-3231.
  25. Hermann R., Monoidal categories and the Gerstenhaber bracket in Hochschild cohomology, Ph.D. Thesis, Bielefeld University, Germany, 2013, arXiv:1403.3597.
  26. Khomenko O., Some applications of Gelfand-Zetlin modules, in Representations of algebras and related topics, Fields Inst. Commun., Vol. 45, Amer. Math. Soc., Providence, RI, 2005, 205-213.
  27. Khomenko O., Mazorchuk V., Structure of modules induced from simple modules with minimal annihilator, Canad. J. Math. 56 (2004), 293-309.
  28. König S., Mazorchuk V., An equivalence of two categories of ${\rm sl}(n,{\mathbb C})$-modules, Algebr. Represent. Theory 5 (2002), 319-329.
  29. Kostant B., On Whittaker vectors and representation theory, Invent. Math. 48 (1978), 101-184.
  30. Krause H., The spectrum of a module category, Mem. Amer. Math. Soc. 149 (2001), x+125 pages.
  31. Mac Lane S., Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
  32. Matlis E., Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511-528.
  33. Mazorchuk V., Tableaux realization of generalized Verma modules, Canad. J. Math. 50 (1998), 816-828.
  34. Mazorchuk V., On Gelfand-Zetlin modules over orthogonal Lie algebras, Algebra Colloq. 8 (2001), 345-360.
  35. Mazorchuk V., Quantum deformation and tableaux realization of simple dense ${\mathfrak{gl}}(n,{\mathbb C})$-modules, J. Algebra Appl. 2 (2003), 1-20.
  36. Mazorchuk V., Ovsienko S., Submodule structure of generalized Verma modules induced from generic Gelfand-Zetlin modules, Algebr. Represent. Theory 1 (1998), 3-26.
  37. Mazorchuk V., Stroppel C., Cuspidal ${\mathfrak{sl}}_n$-modules and deformations of certain Brauer tree algebras, Adv. Math. 228 (2011), 1008-1042, arXiv:1001.2633.
  38. McDowell E., On modules induced from Whittaker modules, J. Algebra 96 (1985), 161-177.
  39. McDowell E., A module induced from a Whittaker module, Proc. Amer. Math. Soc. 118 (1993), 349-354.
  40. Miličić D., Soergel W., The composition series of modules induced from Whittaker modules, Comment. Math. Helv. 72 (1997), 503-520.
  41. Ondrus M., Whittaker modules, central characters, and tensor products for quantum enveloping algebras, Ph.D. Thesis, The University of Wisconsin, Madison, 2004.
  42. Ondrus M., Wiesner E., Whittaker modules for the Virasoro algebra, J. Algebra Appl. 8 (2009), 363-377, arXiv:0805.2686.
  43. Ondrus M., Wiesner E., Whittaker categories for the Virasoro algebra, Comm. Algebra 41 (2013), 3910-3930, arXiv:1108.2698.
  44. Ovsienko S., Finiteness statements for Gelfand-Zetlin modules, in Third International Algebraic Conference in the Ukraine, Inst. of Math., Kiev, 2002, 323-338.
  45. Ovsienko S., Strongly nilpotent matrices and Gelfand-Zetlin modules, Linear Algebra Appl. 365 (2003), 349-367.
  46. Ramirez L.E., Combinatorics of irreducible Gelfand-Tsetlin ${\mathfrak{sl}}(3)$-modules, Algebra Discrete Math. 14 (2012), 276-296.
  47. Roos J.E., Locally Noetherian categories and generalized strictly linearly compact rings. Applications, in Category Theory, Homology Theory and their Applications, II (Battelle Institute Conference, Seattle, Wash., 1968), Springer, Berlin, 1969, 197-277.
  48. Sevostyanov A., Quantum deformation of Whittaker modules and the Toda lattice, Duke Math. J. 105 (2000), 211-238, math.QA/9905128.
  49. Weibel C.A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994.
  50. Zhelobenko D.P., Compact Lie groups and their representations, Nauka, Moscow, 1970.
  51. Zimmermann A., Representation theory. A homological algebra point of view, Algebra and Applications, Vol. 19, Springer International Publishing, Switzerland, 2014.

Previous article  Next article   Contents of Volume 11 (2015)