Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations
To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the integers attached on the vertices are all equal to one. Th...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2010 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2010
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/146522 |
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| Cite this: | Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations / D. Yamakawa // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 31 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862693289184985088 |
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| author | Yamakawa, D. |
| author_facet | Yamakawa, D. |
| citation_txt | Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations / D. Yamakawa // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 31 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the integers attached on the vertices are all equal to one. The construction of reflection functors for quiver varieties are generalized to our case, in which these relate to simple reflections in the Weyl group of some symmetrizable, possibly non-symmetric Kac-Moody algebra. The moduli spaces of meromorphic connections on the rank 2 trivial bundle over the Riemann sphere are described as our manifolds. In our picture, the list of Dynkin diagrams for Painlevé equations is slightly different from (but equivalent to) Okamoto's
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| first_indexed | 2025-12-07T16:19:42Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-146522 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:19:42Z |
| publishDate | 2010 |
| publisher | Інститут математики НАН України |
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| spelling | Yamakawa, D. 2019-02-09T19:50:03Z 2019-02-09T19:50:03Z 2010 Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations / D. Yamakawa // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 31 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D30; 16G20; 20F55; 34M55 DOI:10.3842/SIGMA.2010.087 https://nasplib.isofts.kiev.ua/handle/123456789/146522 To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the integers attached on the vertices are all equal to one. The construction of reflection functors for quiver varieties are generalized to our case, in which these relate to simple reflections in the Weyl group of some symmetrizable, possibly non-symmetric Kac-Moody algebra. The moduli spaces of meromorphic connections on the rank 2 trivial bundle over the Riemann sphere are described as our manifolds. In our picture, the list of Dynkin diagrams for Painlevé equations is slightly different from (but equivalent to) Okamoto's I am grateful to Philip Boalch for stimulating conversations, and to Professor Hiraku Nakajima
 for valuable comments. This work was supported by the grants ANR-08-BLAN-0317-01 of the
 Agence nationale de la recherche and JSPS Grant-in-Aid for Scientific Research (S 19104002). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations Article published earlier |
| spellingShingle | Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations Yamakawa, D. |
| title | Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations |
| title_full | Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations |
| title_fullStr | Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations |
| title_full_unstemmed | Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations |
| title_short | Quiver Varieties with Multiplicities, Weyl Groups of Non-Symmetric Kac-Moody Algebras, and Painlevé Equations |
| title_sort | quiver varieties with multiplicities, weyl groups of non-symmetric kac-moody algebras, and painlevé equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146522 |
| work_keys_str_mv | AT yamakawad quivervarietieswithmultiplicitiesweylgroupsofnonsymmetrickacmoodyalgebrasandpainleveequations |