Tensoring with infinite-dimensional modules in \scr O₀

2007/08/16 by Johan Kåhrström, Kåhrström, Johan
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT

paper · pdf · doi:10.48550/arxiv.0708.2218

arxiv created 2007/08/16 · arxiv updated 2009/12/01

Abstract

We show that the principal block \scr O₀ of the BGG category \scr O for a semisimple Lie algebra \germ 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 \scr O. We study such functors, describe their adjoints and show that they give rise to a natural (co)monad structure on \scr O₀. Furthermore, all this generalises to parabolic subcategories of \scr O₀. As an example, we present some explicit computations for the algebra \germsl₃.

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