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Simple Modules over Lie Algebras
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Simple modules are the elemental components in representation theory for Lie algebras, and numerous mathematicians have worked on their construction and classification over the last century. This thesis consists of an introduction together with four research articles on the subject of simple Lie algebra modules. In the introduction we give a light treatment of the basic structure theory for simple finite dimensional complex Lie algebras and their representations. In particular we give a brief overview of the most well-known classes of Lie algebra modules: highest weight modules, cuspidal modules, Gelfand-Zetlin modules, Whittaker modules, and parabolically induced modules.

The four papers contribute to the subject by construction and classification of new classes of Lie algebra modules. The first two papers focus on U(h)-free modules of rank 1 i.e. modules which are free of rank 1 when restricted to the enveloping algebra of the Cartan subalgebra. In Paper I we classify all such modules for the special linear Lie algebras sln+1(C), and we determine which of these modules are simple. For sl2 we also obtain some additional results on tensor product decomposition. Paper II uses the theory of coherent families to obtain a similar classification for U(h)-free modules over the symplectic Lie algebras sp2n(C). We also give a proof that U(h)-free modules do not exist for any other simple finite-dimensional algebras which completes the classification. In Paper III we construct a new large family of simple generalized Whittaker modules over the general linear Lie algebra gl2n(C). This family of modules is parametrized by non-singular nxn-matrices which makes it the second largest known family of gl2n-modules after the Gelfand-Zetlin modules. In Paper IV we obtain a new class of sln+2(C)-modules by applying the techniques of parabolic induction to the U(h)-free sln+1-modules we constructed in Paper I. We determine necessary and sufficient conditions for these parabolically induced modules to be simple.

Place, publisher, year, edition, pages
Uppsala: Department of Mathematics , 2016. , 50 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 94
Keyword [en]
Lie algebra, Representation, Simple module, Non-weight module, Classification, Construction
National Category
Mathematics Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-283061ISBN: 978-91-506-2544-8 (print)OAI: oai:DiVA.org:uu-283061DiVA: diva2:918131
Public defence
2016-06-01, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2016-05-04 Created: 2016-04-10 Last updated: 2016-05-04
List of papers
1. Simple sl(n+1)-module structures on U(h)
Open this publication in new window or tab >>Simple sl(n+1)-module structures on U(h)
2015 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 424, 294-329 p.Article in journal (Refereed) Published
Abstract [en]

We study the category M consisting of U(sl(n+1))-modules whose restriction to U(h) is free of rank 1, in particular we classify isomorphism classes of objects in M and determine their submodule structure. This leads to new sl(n+1)-modules. For n = 1 we also find the central characters and derive an explicit formula for taking tensor product with a simple finite dimensional module. (C) 2014 Elsevier Inc. All rights reserved.

Keyword
Lie algebra, U(h)-free module, Non-weight module
National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-246334 (URN)10.1016/j.jalgebra.2014.09.036 (DOI)000348334000012 ()
Available from: 2015-03-10 Created: 2015-03-05 Last updated: 2017-12-04Bibliographically approved
2. u(h)-free modules and coherent families
Open this publication in new window or tab >>u(h)-free modules and coherent families
2016 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 220, no 4, 1475-1488 p.Article in journal (Refereed) Published
Abstract [en]

We investigate the category of u(h)-free g-modules. Using a functor from this category to the category of coherent families, we show that u(h)-free modules only can exist when g is of type A or C. We then proceed to classify isomorphism classes of u(h)-free modules of rank 1 in type C, which includes an explicit construction of new simple sp(2n)-modules. The classification is then extended to higher ranks via translation functors.

National Category
Mathematics Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-272027 (URN)10.1016/j.jpaa.2015.09.013 (DOI)000366070600013 ()
Available from: 2016-01-13 Created: 2016-01-11 Last updated: 2017-11-30Bibliographically approved
3. A new family of simple gl2n(C)-modules
Open this publication in new window or tab >>A new family of simple gl2n(C)-modules
2016 (English)In: Pacific Journal of Mathematics, ISSN 0030-8730, E-ISSN 1945-5844, Vol. 283, no 1, 1-19 p.Article in journal (Refereed) Published
Abstract [en]

We construct a new family of simple gl2n-modules which depends on n2 generic parameters. Each module in the family is isomorphic to the regular U(gln )-module when restricted the gln -subalgebra naturally embedded into the top-left corner.

Keyword
Representation theory; Lie algebra; Nonweight module; Whittaker module
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-283050 (URN)10.2140/pjm.2016.283.1 (DOI)000378758000001 ()
Available from: 2016-04-09 Created: 2016-04-09 Last updated: 2017-11-30Bibliographically approved
4. Generalized Verma modules over sl(n+2) induced from U(h)-free sl(n+1)-modules
Open this publication in new window or tab >>Generalized Verma modules over sl(n+2) induced from U(h)-free sl(n+1)-modules
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A class of generalized Verma modules over sl(n+2) is constructed fromsl(n+1)-modules which are U(h)-free modules of rank 1. The necessary and sufficient conditions for these sl(n+2)-modules to be simple are determined. This leads to a class of new simple sl(n+2)-modules.

National Category
Algebra and Logic
Identifiers
urn:nbn:se:uu:diva-281922 (URN)
Available from: 2016-03-31 Created: 2016-03-31 Last updated: 2016-03-31

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