Tame and wild matrix problems
Tame and wild matrix problems
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DOI：
10.1007/bfb0088467
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发表时间：
1980
期刊：
影响因子：
--
通讯作者：
J. Drozd
中科院分区：
文献类型：
--
作者：
J. Drozd
在~ 13~ Nazarova和Roiter证明著名的Brauer-Thrall猜想表明，如果A是一个有限维代数在一个代数闭域然后要么A是有限型，即只有有限数量的非同构不可分解的表示，或其表示的分类包括问题的标准形的矩阵就共轭。在最后一种情况下~是严格无界型，即有一个无限数量的层面，每个拥有无限多的非同构不可分解的表示。许多例子（见~ 2-5、10-123等）证明了无限型代数依次分裂成两个不相交类：“驯服”代数，其不可分解表示可以由几个离散参数和一个连续参数化;和”野性”代数，其表示的分类包括关于共轭的矩阵对的标准形的经典未解决的问题。最后一个问题显然是极端困难的;无论如何，正如F~ uown一样，它包括了任何代数的表示的分类。Freislich~ md Donovan [7]提出了术语”驯服”和”wild”的明确定义，并证明了任何无限型代数要么是驯服，要么是wild。我们证明了这个猜想代数上的代数闭域和一个削弱的形式代数上的完美域。正如在~ 3 J中，证明的自然范围是相当广泛的一类”矩阵问题”，特别是包含在代数表示的分类中出现的问题（一种减少分类的方法）。
In~ 13~ Nazarova and Roiter proving the famous Brauer-Thrall conjecture showed that if A is a finite-dimensional algebra over am algebraically closed field then either A is of finite type, ie has only a finite number of non-isomorphic indecomposable representations, or the classification of its representations includes the problem on the canonical form of matrices with respect to conjugacy. In the last case~ is of strictly unbounded type, ie there is an infinite number of dimensions each possessing infinitely many non-isomorphic indecomposable representations. Numerous examples (see~ 2-5, 10-123 etc.) show that algebras of infinite type split in turn into two disjoint classes:" tame" algebras whose indecomposable representations may be parametrized by several discrete and one contimuos parameters and" wild" algebras for which the classification of representations includes the classical unsolved problem on the canonical form of pairs of matrices with respect to conjugacy. The last problem is apparently of extreme difficulty; at all events, as is well F~ uown, it includes the classification of representations of any algebra.Freislich~ md Donovan [7~ proposed an explicit definition of the terms" tame" and" wild" amd conjectured that any algebra of infinite type is either tame or wild. We prove this conjecture for algebras over an algebraically closed field and in a weakened form for algebras over a perfect field. Just as in~ 3J the natural scope for the proof is a rather wide class of" matrix problems" containing in particular the problems appearing in the classification of representations of algebras (a method for reducing the classifi-