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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
Article

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 \(\frak{sl}_3\).

Keywords

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

Mathematics Subject Classifications (2000)

17B10 17B20 17B35 

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden

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