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Degenerations and other partial orders on the space of representations of algebras
Norwegian University of Science and Technology, Faculty of Information Technology, Mathematics and Electrical Engineering, Department of Mathematical Sciences.
2013 (English)MasteroppgaveStudent thesis
Abstract [en]

Let K be a field and¤be an artin K-algebra. Let r epd¤represent the set of all¤-modules with the length equal to a natural number d as a K-vector space. The set of modules r epd¤ is equipped with the action of the general linear group. The corresponding Zariski-topology for algebraically closed field K then induce a partial order on r epd¤, which is called degeneration order and it is denoted by ·deg . Here for M and N, ¤- modules, the notion M ·deg N mean that the orbit of N under the action of general linear group is contained in the closure of the orbit of M under the same group action. Another partial order on r epd¤ first showed by Riedtmann, is the virtual degeneration order, which is denoted by ·vdeg , are given by M ·vdeg N, if there is a ¤-module X such that M © X ·deg N © X. There are known examples where these two partial orders do not coincide. If K is an algebraically closed field, there is a geometric interpretation of these notions. However, there is also a module theoratical interpretation, which can be generalized to the general settings with K a commutative artin ring. Let ¡ be the Kronecker quiver 1â2 and ¤Æ Z2¡ be the path algebra of ¡ over the field Z2 with two elements. In this work all degenerations between isomrphism classes of modules over ¤ of dimension vector (1, 1), (2,2) and (3,3) are determined and the Hasse diagrams of the corresponding partial orders are given.

Place, publisher, year, edition, pages
Institutt for matematiske fag , 2013. , 36 p.
URN: urn:nbn:no:ntnu:diva-21200Local ID: ntnudaim:10258OAI: diva2:632313
Available from: 2013-06-24 Created: 2013-06-24 Last updated: 2013-06-24Bibliographically approved

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