## 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 $n=4$.

## Citation

Hisaaki FUJITA. Yosuke SAKAI. Daniel SIMSON. "Minor degenerations of the full matrix algebra over a field." J. Math. Soc. Japan 59 (3) 763 - 795, July, 2007. https://doi.org/10.2969/jmsj/05930763

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