Algebras and Representation Theory

, Volume 13, Issue 5, pp 561–587 | Cite as

Tensoring with Infinite-Dimensional Modules in \(\mathcal {O}_0\)

  • Johan KåhrströmEmail author


We show that the principal block \(\mathcal {O}_0\) of the BGG category \(\mathcal {O}\) for a semisimple Lie algebra \(\frak g\) acts faithfully on itself via exact endofunctors which preserve tilting modules, via right exact endofunctors which preserve projective modules and via left exact endofunctors which preserve injective modules. The origin of all these functors is tensoring with arbitrary (not necessarily finite-dimensional) modules in the category \(\mathcal {O}\). We study such functors, describe their adjoints and show that they give rise to a natural (co)monad structure on \(\mathcal {O}_0\). Furthermore, all this generalises to parabolic subcategories of \(\mathcal {O}_0\). As an example, we present some explicit computations for the algebra \(\frak{sl}_3\).


Tensor products BGG category \(\mathcal {O}\) 

Mathematics Subject Classifications (2000)

17B10 17B20 17B35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bass, H.: Algebraic K-theory. Benjamin, New York (1968)Google Scholar
  2. 2.
    Bernstein, J.N., Gelfand, S.I.: Tensor products of finite and infinite dimensional representations of semisimple Lie algebras. Compos. Math. 41(2), 245–285 (1980)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bernstein, J.N., Gelfand, I.M., Gelfand, S.I.: A certain category of \(\frak {g}\text{-modules}\). Funkcional. Anal. i Priložen. 10(2), 1–8 (1976)MathSciNetGoogle Scholar
  4. 4.
    Fiebig, P.: Centers and translation functors for the category \(\mathcal {O}\) over Kac–Moody algebras. Math. Z. 243(4), 689–717 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Irving, R.S.: Projective modules in the category \(\mathcal {O}_S\): Self duality. Trans. Amer. Math. Soc. 291(2), 701–732 (1985)MathSciNetGoogle Scholar
  6. 6.
    Jantzen, J.C.: Moduln mit einem Höchsten Gewicht. Lecture Notes in Mathematics, vol. 750. Springer-Verlag, Berlin (1979)zbMATHGoogle Scholar
  7. 7.
    Lepowsky, J.: A generalization of the Bernstein-Gelfand-Gelfand resolution. J. Algebra 49, 496–511 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Mac Lane, S.: Categories for the Working Mathematician. Springer, New York (1998)zbMATHGoogle Scholar
  9. 9.
    Neidhardt, W.: A translation principle for Kac–Moody algebras. Proc. Amer. Math. Soc. 100(3), 395–400 (1987)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Rocha-Caridi, A.: Splitting criteria for \(\frak {g}\text{-modules}\) induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite dimensional, irreducible \(\frak {g}\text{-module}\). Trans. Amer. Math. Soc. 262(2), 335–366 (1980), DecemberzbMATHMathSciNetGoogle Scholar
  11. 11.
    Rocha-Caridi, A., Wallach, N.R.: Projective modules over graded lie algebras. I. Math. Z. 180, 151–177 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ringel, C.M.: The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences. Math. Z. 208, 209–223 (1991)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Weibel, C.A.: An Introduction to Homological Algebra. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden

Personalised recommendations