2015/12/12 by Gábor Elek, Elek, Gabor
Computer Science · #16E50 #16G10 #Advanced Graph Theory Research #Coding theory and cryptography #Complexity and Algorithms in Graphs #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1512.03959
openalex publication_date 2015/12/12 · openalex created_date 2022/09/26 · openalex updated_date 2026/07/28
We present a novel approach to the representation theory of finite\ndimensional algebras motivated by the emerging theory of graph limits.\n We introduce the rank spectrum of a finite dimensional algebra R over a\nfinite field. The elements of the rank spectrum are representations of the\nalgebra into von Neumann regular rank algebras, and two representations are\nconsidered to be equivalent if they induce the same Sylvester rank functions on\nR-matrices.\n Based on this approach, we can divide the finite dimensional algebras into\nthree types: finite, amenable and non-amenable representation types. We prove\nthat string algebras are of amenable representation type, but the wild\nKronecker algebras are not. Here, the amenability of the rank algebras\nassociated to the limit points in the rank spectrum plays a very important\npart.\n We also show that the limit points of finite dimensional representations of\nalgebras of amenable representation type can always be viewed as\nrepresentations of the algebra in the continuous ring invented by John von\nNeumann in the 1930's.\n As an application in algorithm theory, we introduce and study the notion of\ntesting of parameters of modules over finite dimensional algebras, that is\nanalogous to the property testing of bounded degree graphs introduced by\nGoldreich and Ron. We shall see that for string algebras all the reasonable\n(stable) parameters are testable.\n