Minor degenerations of the full matrix algebra over a field

摘要

Given a positive integer n ≥ 2, an arbitrary field K and an n-block qrn = [q~(1)| … |q~(n)] of n x n square matrices q~(1), ..., q~(n) with coefficients in K satisfying certain conditions, we define a multiplication ·q : M_n{K) (directX)_K M_n(K) → M_n (K) on the K-module M_n (K) of all square n x 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~(qμ)(K) is constructed. We also show that if A_q = M_n~q (K) is a Probenius 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.