On wildness of idempotent generated algebras associated with extended Dynkin diagrams
Let \(\Lambda\) denote an extended Dynkin diagram with vertex set \(\Lambda_0=\{0,1,\ldots, n\}\). For a vertex \(i\), denote by \(S(i)\) the set of vertices \(j\) such that there is an edge joining \(i\) and \(j\); one assumes the diagram has a unique vertex \(p\), say \(p=0\), with \(|S(p)|=3\)....
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Lugansk National Taras Shevchenko University
2018
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oai:ojs.admjournal.luguniv.edu.ua:article-9952018-05-15T06:07:40Z On wildness of idempotent generated algebras associated with extended Dynkin diagrams Bondarenko, Vitalij M. idempotent, extended Dynkin diagram, representation, wild type 16G60; 15A21, 46K10, 46L05 Let \(\Lambda\) denote an extended Dynkin diagram with vertex set \(\Lambda_0=\{0,1,\ldots, n\}\). For a vertex \(i\), denote by \(S(i)\) the set of vertices \(j\) such that there is an edge joining \(i\) and \(j\); one assumes the diagram has a unique vertex \(p\), say \(p=0\), with \(|S(p)|=3\). Further, denote by \(\Lambda\setminus 0\) the full subgraph of \(\Lambda\) with vertex set \(\Lambda_0\setminus\{0\}\). Let \(\Delta=(\delta_i\,|\,i\in \Lambda_0)\in \mathbb{Z}^{|\Lambda_0|}\) be an imaginary root of \(\Lambda\), and let \(k\) be a field of arbitrary characteristic (with unit element 1). We prove that if \(\Lambda\) is an extended Dynkin diagram of type \(\tilde{D_4}\), \(\tilde{E_6}\) or \(\tilde{E_7}\), then the \(k\)-algebra \({\cal Q}_k(\Lambda,\Delta)\) with generators \(e_i\), \(i\in\Lambda_0\setminus\{0\}\), and relations \(e_i^2=e_i\), \(e_ie_j=0\) if \(i\) and \(j\ne i\) belong to the same connected component of \(\Lambda\setminus 0\), and \(\sum_{i=1}^n \delta_i\,e_i=\delta_0 1\) has wild representation type. 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/995 Algebra and Discrete Mathematics; Vol 3, No 3 (2004) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/995/524 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
|
| datestamp_date |
2018-05-15T06:07:40Z |
| collection |
OJS |
| language |
English |
| topic |
idempotent extended Dynkin diagram representation wild type 16G60; 15A21 46K10 46L05 |
| spellingShingle |
idempotent extended Dynkin diagram representation wild type 16G60; 15A21 46K10 46L05 Bondarenko, Vitalij M. On wildness of idempotent generated algebras associated with extended Dynkin diagrams |
| topic_facet |
idempotent extended Dynkin diagram representation wild type 16G60; 15A21 46K10 46L05 |
| format |
Article |
| author |
Bondarenko, Vitalij M. |
| author_facet |
Bondarenko, Vitalij M. |
| author_sort |
Bondarenko, Vitalij M. |
| title |
On wildness of idempotent generated algebras associated with extended Dynkin diagrams |
| title_short |
On wildness of idempotent generated algebras associated with extended Dynkin diagrams |
| title_full |
On wildness of idempotent generated algebras associated with extended Dynkin diagrams |
| title_fullStr |
On wildness of idempotent generated algebras associated with extended Dynkin diagrams |
| title_full_unstemmed |
On wildness of idempotent generated algebras associated with extended Dynkin diagrams |
| title_sort |
on wildness of idempotent generated algebras associated with extended dynkin diagrams |
| description |
Let \(\Lambda\) denote an extended Dynkin diagram with vertex set \(\Lambda_0=\{0,1,\ldots, n\}\). For a vertex \(i\), denote by \(S(i)\) the set of vertices \(j\) such that there is an edge joining \(i\) and \(j\); one assumes the diagram has a unique vertex \(p\), say \(p=0\), with \(|S(p)|=3\). Further, denote by \(\Lambda\setminus 0\) the full subgraph of \(\Lambda\) with vertex set \(\Lambda_0\setminus\{0\}\). Let \(\Delta=(\delta_i\,|\,i\in \Lambda_0)\in \mathbb{Z}^{|\Lambda_0|}\) be an imaginary root of \(\Lambda\), and let \(k\) be a field of arbitrary characteristic (with unit element 1). We prove that if \(\Lambda\) is an extended Dynkin diagram of type \(\tilde{D_4}\), \(\tilde{E_6}\) or \(\tilde{E_7}\), then the \(k\)-algebra \({\cal Q}_k(\Lambda,\Delta)\) with generators \(e_i\), \(i\in\Lambda_0\setminus\{0\}\), and relations \(e_i^2=e_i\), \(e_ie_j=0\) if \(i\) and \(j\ne i\) belong to the same connected component of \(\Lambda\setminus 0\), and \(\sum_{i=1}^n \delta_i\,e_i=\delta_0 1\) has wild representation type. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/995 |
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AT bondarenkovitalijm onwildnessofidempotentgeneratedalgebrasassociatedwithextendeddynkindiagrams |
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2025-07-17T10:31:49Z |
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2025-07-17T10:31:49Z |
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