We present a novel quantitative approach to the representation theory of finite dimensional algebras motivated by the emerging theory of graph limits. We introduce the rank spectrum of a finite dimensional algebra R over a countable field. The elements of the rank spectrum are representations of the algebra into von Neumann regular rank algebras, and two representations are considered to be equivalent if they induce the same Sylvester rank functions on R-matrices. Based on this approach, we can divide the finite dimensional algebras into three types: finite, amenable and non-amenable representation types. We prove that string algebras are of amenable representation type, but the wild Kronecker algebras are not. Here, the amenability of the rank algebras associated to the limit points in the rank spectrum plays a very important part. We also show that the limit points of finite dimensional representations of algebras of amenable representation type can always be viewed as representations of the algebra in the continuous ring invented by John von Neumann in the 1930’s. As an application in algorithm theory, we introduce and study the notion of parameter testing of modules over finite dimensional algebras, that is analogous to the testing of bounded degree graphs introduced by Goldreich and Ron. We shall see that for string algebras all the reasonable (stable) parameters are testable.
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机译：我们提出了一种新颖的量化方法，以图极限的新兴理论为动力，对有限维代数的表示理论进行了研究。我们介绍了可数域上的有限维代数R的秩谱。秩谱的元素是代数到冯·诺依曼规则秩代数的表示，如果两个表示在R矩阵上诱导相同的Sylvester秩函数，则认为这两个表示是等效的。基于这种方法，我们可以将有限维代数分为三种类型：有限，可适应和不可适应的表示类型。我们证明字符串代数是可表示形式的，但野生的Kronecker代数不是。在此，与秩谱中的极限点相关的秩代数的适应性起着非常重要的作用。我们还表明，可以接受表示类型的代数的有限维表示的极限点始终可以视为约翰·冯·诺伊曼（John von Neumann）在1930年代发明的连续环中的代数表示。作为算法理论中的一种应用，我们引入并研究了有限维代数上的模块参数测试的概念，这类似于Goldreich和Ron引入的有界图的测试。我们将看到，对于字符串代数，所有合理（稳定）的参数都是可测试的。
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