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Tensor Products on Category O and Kostant's Problem
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
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. , p. 36
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 59
Keywords [en]
Semi-simple Lie algebras, Tensor products, Kostant's problem, Primitive quotients
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-9388ISBN: 978-91-506-2034-4 (print)OAI: oai:DiVA.org:uu-9388DiVA, id: diva2:172846
Public defence
2008-12-11, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15
Opponent
Supervisors
Available from: 2008-11-19 Created: 2008-11-19Bibliographically approved
List of papers
1. Tensoring with infinite-dimensional modules in O_0
Open this publication in new window or tab >>Tensoring with infinite-dimensional modules in O_0
2010 (English)In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 13, no 5, p. 561-587Article 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).

Keywords
Tensor products, BGG category O
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-97776 (URN)10.1007/s10468-009-9137-6 (DOI)000283587300004 ()
Available from: 2008-11-19 Created: 2008-11-19 Last updated: 2017-12-14Bibliographically approved
2. A new approach to Kostant's problem
Open this publication in new window or tab >>A new approach to Kostant's problem
2010 (English)In: Algebra & number theory, ISSN 1937-0652, Vol. 4, no 3, p. 231-254Article in journal (Refereed) Published
Abstract [en]

For every involution w of the symmetric group S-n we establish, in terms of a special canonical quotient of the dominant Verma module associated with w, an effective criterion to verify whether the universal enveloping algebra U(sl(n)) surjects onto the space of all ad-finite linear transformations of the simple highest weight module L(w). An easy sufficient condition derived from this criterion admits a straightforward computational check (using a computer, for example). All this is applied to get some old and many new results, which answer the classical question of Kostant in special cases; in particular we give a complete answer for simple highest weight modules in the regular block of sl(n), n <= 5.

National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-97777 (URN)000276359100001 ()
Available from: 2008-11-19 Created: 2008-11-19 Last updated: 2011-03-01Bibliographically approved
3. Kostant's problem and parabolic subgroups
Open this publication in new window or tab >>Kostant's problem and parabolic subgroups
2010 (English)In: Glasgow Mathematical Journal, ISSN 0017-0895, E-ISSN 1469-509X, Vol. 52, p. 19-32Article in journal (Refereed) Published
Abstract [en]

Let g be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I subset of S let g(I) be the corresponding semi-simple subalgebra of g. Denote by W-I the Weyl group of g(I) and let w(o) and w(o)(I) be the longest elements of W and W-I, respectively In this paper we show that the answer to Kostant's problem, i.e. whether the Universal enveloping algebra subjects onto the space of all ad-finite linear transformations of a given module, is the same for the simple highest weight g(I)-module L-I(x) of highest weight x . 0, x is an element of W-I, as the answer for the simple highest weight g-module L(xw(o)(l)w(o)) of highest weight xw(o)(I)w(o). 0. We also give a new description Of the unique quasi-simple quotient of the Verma module Delta(e) with the same annihilator as L(y), y is an element of W.

National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-97778 (URN)10.1017/S0017089509990127 (DOI)000273383200002 ()
Available from: 2008-11-19 Created: 2008-11-19 Last updated: 2017-12-14Bibliographically approved

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