uu.seUppsala University Publications

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Simple Modules over Lie AlgebrasPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2016 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Department of Mathematics , 2016. , p. 50
##### 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, id: diva2:918131
##### Public defence

2016-06-01, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
##### Opponent

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Available from: 2016-05-04 Created: 2016-04-10 Last updated: 2016-05-04
##### List of papers

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 sl_{n+1}(C), and we determine which of these modules are simple. For sl_{2} 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 sp_{2n}(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 gl_{2n}(C). This family of modules is parametrized by non-singular nxn-matrices which makes it the second largest known family of gl_{2n}-modules after the Gelfand-Zetlin modules. In Paper IV we obtain a new class of sl_{n+2}(C)-modules by applying the techniques of parabolic induction to the U(h)-free sl_{n+1}-modules we constructed in Paper I. We determine necessary and sufficient conditions for these parabolically induced modules to be simple.

1. Simple sl(n+1)-module structures on U(h)$(function(){PrimeFaces.cw("OverlayPanel","overlay794100",{id:"formSmash:j_idt505:0:j_idt509",widgetVar:"overlay794100",target:"formSmash:j_idt505:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. u(h)-free modules and coherent families$(function(){PrimeFaces.cw("OverlayPanel","overlay893893",{id:"formSmash:j_idt505:1:j_idt509",widgetVar:"overlay893893",target:"formSmash:j_idt505:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. A new family of simple gl_{2n}(C)-modules$(function(){PrimeFaces.cw("OverlayPanel","overlay918102",{id:"formSmash:j_idt505:2:j_idt509",widgetVar:"overlay918102",target:"formSmash:j_idt505:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Generalized Verma modules over sl(n+2) induced from U(h)-free sl(n+1)-modules$(function(){PrimeFaces.cw("OverlayPanel","overlay915983",{id:"formSmash:j_idt505:3:j_idt509",widgetVar:"overlay915983",target:"formSmash:j_idt505:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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