Biserial minor degenerations of matrix algebras over a field

Abstract

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.