Journal article
The geometry of finite dimensional algebras with vanishing radical square
Journal of algebra, Vol.425, pp.146-178
03/01/2015
DOI: 10.1016/j.jalgebra.2014.11.010
Abstract
Let Λ be a basic finite dimensional algebra over an algebraically closed field, with the property that the square of the Jacobson radical J vanishes. We determine the irreducible components of the module variety Repd(Λ) for any dimension vector d. Our description leads to a count of the components in terms of the underlying Gabriel quiver. A closed formula for the number of components when Λ is local extends existing counts for the two-loop quiver to quivers with arbitrary finite sets of loops. For any algebra Λ with J2=0, our criteria for identifying the components of Repd(Λ) permit us to characterize the modules parametrized by the individual irreducible components. Focusing on such a component, we explore generic properties of the corresponding modules by establishing a geometric bridge between the algebras with zero radical square on the one hand and their stably equivalent hereditary counterparts on the other. The bridge links certain closed subvarieties of Grassmannians parametrizing the modules with fixed top over the two types of algebras. By way of this connection, we transfer results of Kac and Schofield from the hereditary case to algebras of Loewy length 2. Finally, we use the transit of information to show that any algebra of Loewy length 2 which enjoys the dense orbit property in the sense of Chindris, Kinser and Weyman has finite representation type.
Details
- Title: Subtitle
- The geometry of finite dimensional algebras with vanishing radical square
- Creators
- Frauke M Bleher - Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419, United StatesTed Chinburg - Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, United StatesBirge Huisgen-Zimmermann - Department of Mathematics, University of California, Santa Barbara, CA 93106-3080, United States
- Resource Type
- Journal article
- Publication Details
- Journal of algebra, Vol.425, pp.146-178
- DOI
- 10.1016/j.jalgebra.2014.11.010
- ISSN
- 0021-8693
- eISSN
- 1090-266X
- Publisher
- Elsevier Inc
- Grant note
- DMS 0932078 / NSF (http://dx.doi.org/10.13039/100000001) H98230-11-1-0131 / NSA DMS 0500961 / NSF (http://dx.doi.org/10.13039/100000001) DMS 110355 / NSF (http://dx.doi.org/10.13039/100000001)
- Language
- English
- Date published
- 03/01/2015
- Academic Unit
- Mathematics
- Record Identifier
- 9983985800802771
Metrics
18 Record Views