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Kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation
We consider the algebras \(e_i \Pi^\lambda(Q) e_i\), where \(\Pi^\lambda(Q)\) is the deformed preprojective algebra of weight \(\lambda\) and \(i\) is some vertex of \(Q\), in the case where \(Q\) is an extended Dynkin diagram and \(\lambda\) lies on the hyperplane orthogonal to the minimal positive...
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Lugansk National Taras Shevchenko University
2018
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oai:ojs.admjournal.luguniv.edu.ua:article-10022018-05-15T06:07:40Z Kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation Mellit, Anton We consider the algebras \(e_i \Pi^\lambda(Q) e_i\), where \(\Pi^\lambda(Q)\) is the deformed preprojective algebra of weight \(\lambda\) and \(i\) is some vertex of \(Q\), in the case where \(Q\) is an extended Dynkin diagram and \(\lambda\) lies on the hyperplane orthogonal to the minimal positive imaginary root \(\delta\). We prove that the center of \(e_i \Pi^\lambda(Q) e_i\) is isomorphic to \(\mathcal{O}^\lambda(Q)\), a deformation of the coordinate ring of the Kleinian singularity that corresponds to \(Q\). We also find a minimal \(k\) for which a standard identity of degree \(k\) holds in \(e_i \Pi^\lambda(Q) e_i\). We prove that the algebras \(A_{P_1,\dots,P_n;\mu} = \mathbb{C}\langle x_1, \dots, x_n | P_i(x_i)=0, \sum_{i=1}^n x_i = \mu e\rangle\) make a special case of the algebras \(e_c \Pi^\lambda(Q) e_c\) for star-like quivers \(Q\) with the origin \(c\). Lugansk National Taras Shevchenko University 2018-05-15 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1002 Algebra and Discrete Mathematics; Vol 3, No 3 (2004) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1002/531 Copyright (c) 2018 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
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English |
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| spellingShingle |
Mellit, Anton Kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation |
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| format |
Article |
| author |
Mellit, Anton |
| author_facet |
Mellit, Anton |
| author_sort |
Mellit, Anton |
| title |
Kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation |
| title_short |
Kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation |
| title_full |
Kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation |
| title_fullStr |
Kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation |
| title_full_unstemmed |
Kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation |
| title_sort |
kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation |
| description |
We consider the algebras \(e_i \Pi^\lambda(Q) e_i\), where \(\Pi^\lambda(Q)\) is the deformed preprojective algebra of weight \(\lambda\) and \(i\) is some vertex of \(Q\), in the case where \(Q\) is an extended Dynkin diagram and \(\lambda\) lies on the hyperplane orthogonal to the minimal positive imaginary root \(\delta\). We prove that the center of \(e_i \Pi^\lambda(Q) e_i\) is isomorphic to \(\mathcal{O}^\lambda(Q)\), a deformation of the coordinate ring of the Kleinian singularity that corresponds to \(Q\). We also find a minimal \(k\) for which a standard identity of degree \(k\) holds in \(e_i \Pi^\lambda(Q) e_i\). We prove that the algebras \(A_{P_1,\dots,P_n;\mu} = \mathbb{C}\langle x_1, \dots, x_n | P_i(x_i)=0, \sum_{i=1}^n x_i = \mu e\rangle\) make a special case of the algebras \(e_c \Pi^\lambda(Q) e_c\) for star-like quivers \(Q\) with the origin \(c\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1002 |
| work_keys_str_mv |
AT mellitanton kleiniansingularitiesandalgebrasgeneratedbyelementsthathavegivenspectraandsatisfyascalarsumrelation |
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2024-04-12T06:27:34Z |
| last_indexed |
2024-04-12T06:27:34Z |
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1796109242900414464 |