Minor degenerations of the full matrix algebra over a field

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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 : Mn(K)$¥otimes$K Mn(K)→Mn(K) on the K-module Mn(K) of all square n×n matrices with coefficients in K in such a way that ·q defines a K-algebra structure on Mn(K). We denote it by Mqn(K), and we call it a minor q-degeneration of the full matrix K-algebra Mn(K). The class of minor degenerations of the algebra Mn(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 Mqn(K) is described and conditions for q to be Mqn(K) a Frobenius algebra are given. In case K is an infinite field, for each n≥4 a one-parameter K-algebraic family {Cμ}μ∈K* of basic pairwise non-isomorphic Frobenius K-algebras of the form Cμ=Mqμn(K) is constructed. We also show that if Aq=Mqn(K) is a Frobenius algebra such that J(Aq)3=0, then Aq is representation-finite if and only if n=3, and Aq is tame representation-infinite if and only if n=4.
社団法人日本数学会の論文
2007-07-01