Abstract
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 \(\frak{sl}_3\).
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References
Bass, H.: Algebraic K-theory. Benjamin, New York (1968)
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)
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)
Fiebig, P.: Centers and translation functors for the category \(\mathcal {O}\) over Kac–Moody algebras. Math. Z. 243(4), 689–717 (2003)
Irving, R.S.: Projective modules in the category \(\mathcal {O}_S\): Self duality. Trans. Amer. Math. Soc. 291(2), 701–732 (1985)
Jantzen, J.C.: Moduln mit einem Höchsten Gewicht. Lecture Notes in Mathematics, vol. 750. Springer-Verlag, Berlin (1979)
Lepowsky, J.: A generalization of the Bernstein-Gelfand-Gelfand resolution. J. Algebra 49, 496–511 (1977)
Mac Lane, S.: Categories for the Working Mathematician. Springer, New York (1998)
Neidhardt, W.: A translation principle for Kac–Moody algebras. Proc. Amer. Math. Soc. 100(3), 395–400 (1987)
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), December
Rocha-Caridi, A., Wallach, N.R.: Projective modules over graded lie algebras. I. Math. Z. 180, 151–177 (1982)
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)
Weibel, C.A.: An Introduction to Homological Algebra. Cambridge University Press, Cambridge (1995)
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Kåhrström, J. Tensoring with Infinite-Dimensional Modules in \(\mathcal {O}_0\) . Algebr Represent Theor 13, 561–587 (2010). https://doi.org/10.1007/s10468-009-9137-6
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DOI: https://doi.org/10.1007/s10468-009-9137-6