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Algebra and Discrete Mathematics, 2010, том 9, выпуск 2, страницы 127–139
(Mi adm34)
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RESEARCH ARTICLE
Biserial minor degenerations of matrix algebras over a field
Anna Włodarska Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, 87-100 Toruń, Poland
Аннотация:
Let $n\geq 2$ be a positive integer, $K$ an arbitrary field, and $q=[q^{(1)}|\dots|q^{(n)}]$ an $n$-block matrix of $n\times n$ square matrices $q^{(1)},\dots,q^{(n)}$ 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–Saka—Simson [7]. A characterisation of all block matrices $q=[q^{(1)}|\dots|q^{(n)}]$ 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.
Ключевые слова:
right special biserial algebra, biserial algebra, Gabriel quiver.
Поступила в редакцию: 09.03.2010 Исправленный вариант: 14.10.2010
Образец цитирования:
Anna Włodarska, “Biserial minor degenerations of matrix algebras over a field”, Algebra Discrete Math., 9:2 (2010), 127–139
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/adm34 https://www.mathnet.ru/rus/adm/v9/i2/p127
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