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Tensoring with infinite-dimensional modules in O_0
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
2010 (English)In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 13, no 5, 561-587 p.Article in journal (Refereed) Published
Abstract [en]

We show that the principal block O-0 of the BGG category O for a semi-simple Lie algebra 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 O. We study such functors, describe their adjoints and show that they give rise to a natural (co) monad structure on O-0. Furthermore, all this generalises to parabolic subcategories of O-0. As an example, we present some explicit computations for the algebra sl(3).

Place, publisher, year, edition, pages
2010. Vol. 13, no 5, 561-587 p.
Keyword [en]
Tensor products, BGG category O
National Category
URN: urn:nbn:se:uu:diva-97776DOI: 10.1007/s10468-009-9137-6ISI: 000283587300004OAI: oai:DiVA.org:uu-97776DiVA: diva2:172843
Available from: 2008-11-19 Created: 2008-11-19 Last updated: 2011-03-01Bibliographically approved
In thesis
1. Tensor Products on Category O and Kostant's Problem
Open this publication in new window or tab >>Tensor Products on Category O and Kostant's Problem
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of a summary and three papers, concerning some aspects of representation theory for complex finite dimensional semi-simple Lie algebras with focus on the BGG-category O.

Paper I is motivated by the many useful properties of functors on category O given by tensoring with finite dimensional modules, such as projective functors and translation functors. We study properties of functors on O given by tensoring with arbitrary (possibly infinite dimensional) modules. Such functors give rise to a faithful action of O on itself via exact functors which preserve tilting modules, via right exact functors which preserve projective modules, and via left exact functors which preserve injective modules.

Papers II and III both deal with Kostant's problem. In Paper II we establish an effective criterion equivalent to the answer to Kostant's problem for simple highest weight modules, in the case where the Lie algebra is of type A. Using this, we derive some old and new results which answer Kostant's problem in special cases. An easy sufficient condition derived from this criterion using Kazhdan-Lusztig combinatorics allows for a straightforward computational check using a computer, by which we get a complete answer for simple highest weight modules in the principal block of O for algebras of rank less than 5.

In Paper III we relate the answer to Kostant's problem for certain modules to the answer to Kostant's problem for a module over a subalgebra. We also give a new description of a certain quotient of the dominant Verma module, which allows us to give a bound on the multiplicities of simple composition factors of primitive quotients of the universal enveloping algebra.

Place, publisher, year, edition, pages
Uppsala: Universitetsbiblioteket, 2008. 36 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 59
Semi-simple Lie algebras, Tensor products, Kostant's problem, Primitive quotients
National Category
urn:nbn:se:uu:diva-9388 (URN)978-91-506-2034-4 (ISBN)
Public defence
2008-12-11, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15
Available from: 2008-11-19 Created: 2008-11-19Bibliographically approved

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