Henning Krause
Given an abelian length category A, the Gabriel-Roiter measure with respect to a length function l is characterized as a universal morphism ind(A)-->P of partially ordered sets. The map is defined on the isomorphism classes of indecomposable objects of A and is a suitable refinement of the length function l.
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These notes give an introduction to the Gabriel-Roiter measure of a finite dimensional algebra. They are based on a series of four lectures at the "Advanced School and Conference on Representation Theory and Related Topics" in Trieste (ICTP, January 2006).
Let $Λ$ be an artin algebra. The aim of this paper is to outline a strong relationship between the Gabriel-Roiter inclusions and the Auslander-Reiten theory. If $X$ is a Gabriel-Roiter submodule of $Y,$ then $Y$ is shown to be a factor module of an indecomposable module $M$ such that there exists an irreducible monomorphism $X \to M$. We also will prove that the monomorphisms in a homogeneous tube are Gabriel-Roiter inclusions, provided the the tube contains a module whose endomorphism ring is a division ring.
Let $Q$ be an $n$-Kronecker quiver, i.e., a quiver with two vertices, labeled by 1 and 2, and $n$ arrows from 2 to 1. We show that there are infinitely many Gabriel-Roiter measures admitting no direct predecessors.
Let $Q$ be a wild $n$-Kronecker quiver, i.e., a quiver with two vertices, labeled by 1 and 2, and $n\geq 3$ arrows from 2 to 1. The indecomposable regular modules with preprojective Gabriel-Roiter submodules, in particular, those $τ^{-i}X$ with $\udim X=(1,c)$ for $i\geq 0$ and some $1\leq c\leq n-1$ will be studied. It will be shown that for each $i\geq 0$ the irreducible monomorphisms starting with $τ^{-i}X$ give rise to a sequence of Gabriel-Roiter inclusions, and moreover, the Gabriel-Roiter measures of those produce a sequence of direct successors. In particular, there are infinitely many…
Let $Q$ be a tame quiver of type $\widetilde{\mathbb{A}}_n$ and $\Rep(Q)$ the category of finite dimensional representations over an algebraically closed field. A representation is simply called a module. It will be shown that a regular string module has, up to isomorphism, at most two Gabriel-Roiter submodules. The quivers $Q$ with sink-source orientations will be characterized as those, whose central parts do not contain preinjective modules. It will also be shown that there are only finitely many (central) Gabriel-Roiter measures admitting no direct predecessors. This fact will be generaliz…
Let $Q$ be the 3-Kronecker quiver, i.e., $Q$ has two vertices, labeled by 1 and 2, and three arrows from 2 to 1. Fix an algebraically closed field $k$. Let $\mathcal{C}$ be a regular component of the Auslander-Reiten quiver containing an indecomposable module $X$ with dimension $(1,1)$ or $(2,1)$. Using the properties of the Fibonacci numbers, we show that the Gabriel-Roiter measures of the indecomposable modules in $\mathcal{C}$ are uniquely determined by the dimension vectors. In other words, two indecomposable modules in $\mathcal{C}$ are not isomorphic if and only if their Gabriel-Roiter m…
A famous result by Drozd says that a finite-dimensional representation-infinite algebra is of either tame or wild representation type. But one has to make assumption on the ground field. The Gabriel-Roiter measure might be an alternative approach to extend these concepts of tame and wild to arbitrary artin algebras. In particular, the infiniteness of the number of GR segments, i.e. sequences of Gabriel-Roiter measures which are closed under direct predecessors and successors, might relate to the wildness of artin algebras. As the first step, we are going to study the wild quiver with three ver…
The Gabriel-Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel-Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel-Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel-Roiter measure $μ$ in an abelian length category $\mathcal{A}$, there exists an object $X'$ which depends on $X$ and $μ$, such that $Γ= \operatorname{End}_{\mathcal{A}}(X \oplus X')$ h…
Generalised Gabriel-Roiter measure and thin representations
该研究针对Dynkin与Euclidean箭图，揭示thin表示的Gabriel-Roiter测度与子箭图系统诱导链长函数的等价性，给出求GR滤过的组合方法，证明其域无关性及不可分解滤过的GR滤过性质。
Inclusion preserving maps from modules over an Artin algebra to complete partially ordered sets are studied. This yields a filtration of the Ziegler spectrum which is indexed by all Gabriel-Roiter measures. Another application is a compactness result for the set of subcategories of finitely presented modules that are closed under submodules.
Let k be an arbitrary field and Q an acyclic quiver of tame type. Consider the path algebra kQ and the category of finite dimensional right modules Mod kQ. In the first part of the paper we deduce that the Gabriel-Roiter inclusions in preprojective indecomposables and simple homogeneous regulars (with dimension the minimal radical vector) as well as their Gabriel-Roiter measures are field independent. Using this result we can prove in a more general setting a theorem by Bo Chen which states that the Gabriel-Roiter submodule of a simple homogeneous regular (with dimension the minimal radical ve…
Let $Λ$ be an Artin algebra. A GR segment of $Λ$ is a sequence of GR measures which is closed under direct successors and direct predecessors. The number of the GR segments was conjectured to relate to the representation type of $Λ$. In this paper, let $k$ be an algebraically closed field and $Λ$ be a finite-dimensional hereditary $k$-algebra. We show that $Λ$ admits infinitely many GR segments if and only if $Λ$ is of wild representation type. Thus the finiteness of the number of the GR segments might be an alternative characterization of the tameness of finite dimensional algebras over algeb…
By [R. Bautista, P. Gabriel, A.V Roiter., L. Salmeron, Representation-finite algebras and multiplicative basis. Invent. Math. 81 (1985) 217-285.], a finite-dimensional algebra having finitely many isoclasses of indecomposable representations admits a multiplicative basis. In Sections 4.10-4.12 of [P. Gabriel, A. V. Roiter, Representations of finite-dimensional algebras. Encyclopaedia of Math. Sci., vol. 73, Algebra 8, Springer-Verlag, 1992] an analogous hypothesis was formulated for finitely spaced modules over an aggregate. We prove this conjecture.
The first Brauer-Thrall conjecture for extriangulated length categories
本文在外三角长度范畴中引入关于 $Θ$ 的 Gabriel-Roiter 测度并推广 Gabriel 性质，在技术性条件下证明第一 Brauer-Thrall 猜想。
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