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Tensoring with infinite-dimensional modules in O_0
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra, geometri och logik.
2010 (Engelska)Ingår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 13, nr 5, s. 561-587Artikel i tidskrift (Refereegranskat) 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).

Ort, förlag, år, upplaga, sidor
2010. Vol. 13, nr 5, s. 561-587
Nyckelord [en]
Tensor products, BGG category O
Nationell ämneskategori
Matematik
Identifikatorer
URN: urn:nbn:se:uu:diva-97776DOI: 10.1007/s10468-009-9137-6ISI: 000283587300004OAI: oai:DiVA.org:uu-97776DiVA, id: diva2:172843
Tillgänglig från: 2008-11-19 Skapad: 2008-11-19 Senast uppdaterad: 2017-12-14Bibliografiskt granskad
Ingår i avhandling
1. Tensor Products on Category O and Kostant's Problem
Öppna denna publikation i ny flik eller fönster >>Tensor Products on Category O and Kostant's Problem
2008 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Uppsala: Universitetsbiblioteket, 2008. s. 36
Serie
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 59
Nyckelord
Semi-simple Lie algebras, Tensor products, Kostant's problem, Primitive quotients
Nationell ämneskategori
Matematik
Identifikatorer
urn:nbn:se:uu:diva-9388 (URN)978-91-506-2034-4 (ISBN)
Disputation
2008-12-11, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15
Opponent
Handledare
Tillgänglig från: 2008-11-19 Skapad: 2008-11-19Bibliografiskt granskad

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