Abstract
Given a positive integer , an arbitrary field and an -block of square matrices with coefficients in 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 -module of all square matrices with coefficients in in such a way that defines a -algebra structure on . We denote it by , and we call it a minor -degeneration of the full matrix -algebra . The class of minor degenerations of the algebra and their modules are investigated in the paper by means of the properties of and by applying quivers with relations. The Gabriel quiver of is described and conditions for to be a Frobenius algebra are given. In case is an infinite field, for each a one-parameter -algebraic family of basic pairwise non-isomorphic Frobenius -algebras of the form is constructed. We also show that if is a Frobenius algebra such that , then is representation-finite if and only if , and is tame representation-infinite if and only if .
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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