uu.seUppsala University Publications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
A new approach to Kostant's problem
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.
2010 (English)In: Algebra & number theory, ISSN 1937-0652, Vol. 4, no 3, 231-254 p.Article 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.

Place, publisher, year, edition, pages
2010. Vol. 4, no 3, 231-254 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-97777ISI: 000276359100001OAI: oai:DiVA.org:uu-97777DiVA: diva2:172844
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.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 59
Keyword
Semi-simple Lie algebras, Tensor products, Kostant's problem, Primitive quotients
National Category
Mathematics
Identifiers
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
Opponent
Supervisors
Available from: 2008-11-19 Created: 2008-11-19Bibliographically approved

Open Access in DiVA

No full text

By organisation
Algebra, Geometry and Logic
Mathematics

Search outside of DiVA

GoogleGoogle Scholar

urn-nbn

Altmetric score

urn-nbn
Total: 469 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf