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 $._q : \mathbf{M}_n(K) \otimes_K \mathbf{M}_n(K) \rightarrow \mathbf{M}_n(K)$ 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 n q ( 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 n q ( K ) is described and conditions for q to be M n q ( 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 n q μ ( K ) is constructed. We also show that if A q = M n q ( 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 .