Biserial minor degenerations of matrix algebras over a field
Let \(n\geq 2\) be a positive integer, \(K\) an arbitrary field, and \( q = [ q ^{(1)}| \ldots | q ^{(n)}]\) an \(n\)-block matrix of \(n\times n\) square matrices \( q ^{(1)}, \ldots, q ^{(n)}\) with coefficients in \(K\) satisfying the conditions (C1) and (C2) listed in the introduction. We study...
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/636 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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admjournalluguniveduua-article-6362018-04-04T09:11:25Z Biserial minor degenerations of matrix algebras over a field Wlodarska, Anna right special biserial algebra, biserial algebra, Gabriel quiver 16G10, 16G60, 14R20, 16S80 Let \(n\geq 2\) be a positive integer, \(K\) an arbitrary field, and \( q = [ q ^{(1)}| \ldots | q ^{(n)}]\) an \(n\)-block matrix of \(n\times n\) square matrices \( q ^{(1)}, \ldots, q ^{(n)}\) with coefficients in \(K\) satisfying the conditions (C1) and (C2) listed in the introduction. We study minor degenerations \(\mathbb{M}^q_n(K)\) of the full matrix algebra \(\mathbb{M}_n(K)\) in the sense of Fujita-Sakai-Simson [7]. A characterisation of all block matrices \( q = [ q ^{(1)}| \ldots | q ^{(n)}]\) such that the algebra \(\mathbb{M}^q_n(K)\) is basic and right biserial is given in the paper. We also prove that a basic algebra \(\mathbb{M}^q_n(K)\) is right biserial if and only if \(\mathbb{M}^q_n(K)\) is right special biserial. It is also shown that the \(K\)-dimensions of the left socle of \(\mathbb{M}^q_n(K)\) and of the right socle of \(\mathbb{M}^q_n(K)\) coincide, in case \(\mathbb{M}^q_n(K)\) is basic and biserial. Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/636 Algebra and Discrete Mathematics; Vol 9, No 2 (2010) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/636/170 Copyright (c) 2018 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
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2018-04-04T09:11:25Z |
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OJS |
| language |
English |
| topic |
right special biserial algebra biserial algebra Gabriel quiver 16G10 16G60 14R20 16S80 |
| spellingShingle |
right special biserial algebra biserial algebra Gabriel quiver 16G10 16G60 14R20 16S80 Wlodarska, Anna Biserial minor degenerations of matrix algebras over a field |
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right special biserial algebra biserial algebra Gabriel quiver 16G10 16G60 14R20 16S80 |
| format |
Article |
| author |
Wlodarska, Anna |
| author_facet |
Wlodarska, Anna |
| author_sort |
Wlodarska, Anna |
| title |
Biserial minor degenerations of matrix algebras over a field |
| title_short |
Biserial minor degenerations of matrix algebras over a field |
| title_full |
Biserial minor degenerations of matrix algebras over a field |
| title_fullStr |
Biserial minor degenerations of matrix algebras over a field |
| title_full_unstemmed |
Biserial minor degenerations of matrix algebras over a field |
| title_sort |
biserial minor degenerations of matrix algebras over a field |
| description |
Let \(n\geq 2\) be a positive integer, \(K\) an arbitrary field, and \( q = [ q ^{(1)}| \ldots | q ^{(n)}]\) an \(n\)-block matrix of \(n\times n\) square matrices \( q ^{(1)}, \ldots, q ^{(n)}\) with coefficients in \(K\) satisfying the conditions (C1) and (C2) listed in the introduction. We study minor degenerations \(\mathbb{M}^q_n(K)\) of the full matrix algebra \(\mathbb{M}_n(K)\) in the sense of Fujita-Sakai-Simson [7]. A characterisation of all block matrices \( q = [ q ^{(1)}| \ldots | q ^{(n)}]\) such that the algebra \(\mathbb{M}^q_n(K)\) is basic and right biserial is given in the paper. We also prove that a basic algebra \(\mathbb{M}^q_n(K)\) is right biserial if and only if \(\mathbb{M}^q_n(K)\) is right special biserial. It is also shown that the \(K\)-dimensions of the left socle of \(\mathbb{M}^q_n(K)\) and of the right socle of \(\mathbb{M}^q_n(K)\) coincide, in case \(\mathbb{M}^q_n(K)\) is basic and biserial. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/636 |
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2025-12-02T15:43:02Z |
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