Minor degenerations of the full matrix algebra over a field

Abstract

Given a positive integer $n\ge 2$, an arbitrary field $K$ and an $n$-block $q=\left[q^{\left(1\right)}\right|\cdots \left|q^{\left(n\right)}\right]$ of $n\times n$ square matrices $q^{\left(1\right)},\dots ,q^{\left(n\right)}$ with coefficients in $K$ satisfying certain conditions, we define a multiplication $._q : \mathbf{M}_n(K) \otimes_K \mathbf{M}_n(K) \rightarrow \mathbf{M}_n(K)$ on the $K$-module ${\mathbf{M}}_n\left(K\right)$ of all square $n\times n$ matrices with coefficients in $K$ in such a way that ${\cdot }_q$ defines a $K$-algebra structure on ${\mathbf{M}}_n\left(K\right)$. We denote it by ${\mathbf{M}}_n^q\left(K\right)$, and we call it a minor $q$-degeneration of the full matrix $K$-algebra ${\mathbf{M}}_n\left(K\right)$. The class of minor degenerations of the algebra ${\mathbf{M}}_n\left(K\right)$ and their modules are investigated in the paper by means of the properties of $q$ and by applying quivers with relations. The Gabriel quiver of ${\mathbf{M}}_n^q\left(K\right)$ is described and conditions for $q$ to be ${\mathbf{M}}_n^q\left(K\right)$ a Frobenius algebra are given. In case $K$ is an infinite field, for each $n\ge 4$ a one-parameter $K$-algebraic family $\{C_{\mu }{\}}_{\mu \in K^{*}}$ of basic pairwise non-isomorphic Frobenius $K$-algebras of the form $C_{\mu }={\mathbf{M}}_n^{q_{\mu }}\left(K\right)$ is constructed. We also show that if $A_q={\mathbf{M}}_n^q\left(K\right)$ is a Frobenius algebra such that $J(A_q{)}^3=0$, then $A_q$ is representation-finite if and only if $n=3$, and $A_q$ is tame representation-infinite if and only if $n=4$.