Quivers and the Euclidean algebra (Extended abstract)
We show that the category of representations of the Euclidean group $E(2)$ is equivalent to the category of representations of the preprojective algebra of the quiver of type $A_{\infty}$. Furthermore, we consider the moduli space of $E(2)$-modules along with a set of generators. We show that these...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2008-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/3603/pdf |
| _version_ | 1797270352131260416 |
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| author | Alistair Savage |
| author_facet | Alistair Savage |
| author_sort | Alistair Savage |
| collection | DOAJ |
| description | We show that the category of representations of the Euclidean group $E(2)$ is equivalent to the category of representations of the preprojective algebra of the quiver of type $A_{\infty}$. Furthermore, we consider the moduli space of $E(2)$-modules along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. These identifications allow us to draw on known results about preprojective algebras and quiver varieties to prove various statements about representations of $E(2)$. In particular, we show that $E(2)$ has wild representation type but that if we impose certain combinatorial restrictions on the weight decompositions of a representation, we obtain only a finite number of indecomposable representations. |
| first_indexed | 2024-04-25T02:02:54Z |
| format | Article |
| id | doaj.art-6704880d3632448e8ca518a921105e0f |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:54Z |
| publishDate | 2008-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-6704880d3632448e8ca518a921105e0f2024-03-07T14:38:07ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502008-01-01DMTCS Proceedings vol. AJ,...Proceedings10.46298/dmtcs.36033603Quivers and the Euclidean algebra (Extended abstract)Alistair Savage0University of Ottawa [Ottawa]We show that the category of representations of the Euclidean group $E(2)$ is equivalent to the category of representations of the preprojective algebra of the quiver of type $A_{\infty}$. Furthermore, we consider the moduli space of $E(2)$-modules along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. These identifications allow us to draw on known results about preprojective algebras and quiver varieties to prove various statements about representations of $E(2)$. In particular, we show that $E(2)$ has wild representation type but that if we impose certain combinatorial restrictions on the weight decompositions of a representation, we obtain only a finite number of indecomposable representations.https://dmtcs.episciences.org/3603/pdfquiverquiver varietydirected grapheuclidean algebraeuclidean grouppreprojective algebra[math.math-co] mathematics [math]/combinatorics [math.co][info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Alistair Savage Quivers and the Euclidean algebra (Extended abstract) Discrete Mathematics & Theoretical Computer Science quiver quiver variety directed graph euclidean algebra euclidean group preprojective algebra [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Quivers and the Euclidean algebra (Extended abstract) |
| title_full | Quivers and the Euclidean algebra (Extended abstract) |
| title_fullStr | Quivers and the Euclidean algebra (Extended abstract) |
| title_full_unstemmed | Quivers and the Euclidean algebra (Extended abstract) |
| title_short | Quivers and the Euclidean algebra (Extended abstract) |
| title_sort | quivers and the euclidean algebra extended abstract |
| topic | quiver quiver variety directed graph euclidean algebra euclidean group preprojective algebra [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/3603/pdf |
| work_keys_str_mv | AT alistairsavage quiversandtheeuclideanalgebraextendedabstract |