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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2010, Volume 9, Issue 2, Pages 127–139 (Mi adm34)

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

Abstract: Let $n\geq 2$ be a positive integer, $K$ an arbitrary field, and $q=[q^{(1)}|…|q^{(n)}]$ an $n$-block matrix of $n\times n$ square matrices $q^{(1)},…,q^{(n)}$ 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 $q=[q^{(1)}|…|q^{(n)}]$ 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.

Keywords: right special biserial algebra, biserial algebra, Gabriel quiver.

Full text: PDF file (262 kB)

Bibliographic databases:
MSC: 16G10, 16G60, 14R20, 16S80
Revised: 14.10.2010
Language:

Citation: Anna Włodarska, “Biserial minor degenerations of matrix algebras over a field”, Algebra Discrete Math., 9:2 (2010), 127–139

Citation in format AMSBIB
\Bibitem{Wlo10} \by Anna W{\l}odarska \paper Biserial minor degenerations of matrix algebras over a~field \jour Algebra Discrete Math. \yr 2010 \vol 9 \issue 2 \pages 127--139 \mathnet{http://mi.mathnet.ru/adm34} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2808786} \zmath{https://zbmath.org/?q=an:1209.16021}