Infinite dimensional representations of finite dimensional algebras and amenability
Infinite dimensional representations of finite dimensional algebras and amenability
复制标题
有限维代数的无限维表示和适用性
DOI：
10.1007/s00208-017-1552-0
复制
发表时间：
2015
影响因子：
1.4
通讯作者：
G. Elek
中科院分区：
文献类型：
--
作者：
G. Elek
在新兴的图极限理论的推动下，我们提出了一种新颖的有限维代数表示论定量方法。我们引入可数域上的有限维代数 R 的秩谱。秩谱的元素是代数到冯·诺依曼正则秩代数的表示，并且如果两个表示在 R 矩阵上导出相同的西尔维斯特秩函数，则认为它们是等价的。基于这种方法，我们可以将有限维代数分为三种类型：有限的、顺从的和非顺从的表示类型。我们证明了弦代数是合适的表示类型，但狂野克罗内克代数却不是。在这里，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，适合表示类型的代数的有限维表示的极限点始终可以被视为约翰·冯·诺依曼在 1930 年代发明的连续环中的代数表示。作为算法理论中的应用，我们介绍并研究了有限维代数上模的参数测试的概念，这类似于 Goldreich 和 Ron 引入的有界度图的测试。我们将看到，对于弦代数，所有合理（稳定）的参数都是可测试的。
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.
代数和模的派生本地化
DOI：
--
发表时间：
2015
期刊：
--
影响因子：
--
作者：
C. Braun
通讯作者：
C. Braun