Biserial minor degenerations of matrix algebras over a field

Anna Włodarska

Algebra and discrete mathematics, Tome 9 (2010) no. 2, pp. 127-139

Résumé

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.

Détail

Keywords: right special biserial algebra, biserial algebra
Mots-clés : Gabriel quiver.