Minor degenerations of the full matrix algebra over a field

Abstract

Given a positive integer n≥2, an arbitrary field K and an n-block q=[q⁽¹⁾|…|q⁽ⁿ⁾] of n×n square matrices q⁽¹⁾,…,q⁽ⁿ⁾ with coefficients in K satisfying certain conditions, we define a multiplication ·_q : M_n(K)$¥otimes$_K M_n(K)→M_n(K) on the K-module M_n(K) of all square n×n matrices with coefficients in K in such a way that ·_q defines a K-algebra structure on M_n(K). We denote it by M^q_n(K), and we call it a minor q-degeneration of the full matrix K-algebra M_n(K). The class of minor degenerations of the algebra M_n(K) 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 M^q_n(K) is described and conditions for q to be M^q_n(K) a Frobenius algebra are given. In case K is an infinite field, for each n≥4 a one-parameter K-algebraic family {C_μ}_μ∈K* of basic pairwise non-isomorphic Frobenius K-algebras of the form C_μ=M^qμ_n(K) is constructed. We also show that if A_q=M^q_n(K) 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.