共查询到20条相似文献,搜索用时 15 毫秒
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Liu and Paquette defined a class of artin algebras, more general than the standardly stratified ones, called quasi-stratified algebras. Not only is the Cartan Determinant Conjecture (CDC) true for these algebras, so is its converse. This article shows that this class of algebras is preserved under “pruning” sources and sinks from the left quiver. It compares the classes of quasi-stratified and left serial algebras, as well as quasi-stratified and gentle algebras. Holm has shown that the CDC holds for gentle algebras; the converse is also established. It is shown when a Yamagata family of algebras of large finite global dimension yield quasi-stratified ones and constructs quasi-stratified elementary algebras from smaller ones. 相似文献
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The motivation of this paper is to study the natural quiver of an artinian algebra, a new kind of quivers, as a tool independing
upon the associated basic algebra. In Li (J Aust Math Soc 83:385–416, 2007), the notion of the natural quiver of an artinian algebra was introduced and then was used to generalize the Gabriel theorem
for non-basic artinian algebras splitting over radicals and non-basic finite dimensional algebras with 2-nilpotent radicals
via pseudo path algebras and generalized path algebras respectively. In this paper, firstly we consider the relationship between
the natural quiver and the ordinary quiver of a finite dimensional algebra. Secondly, the generalized Gabriel theorem is obtained
for radical-graded artinian algebras. Moreover, Gabriel-type algebras are introduced to outline those artinian algebras satisfying
the generalized Gabriel theorem here and in Li (J Aust Math Soc 83:385–416, 2007). For such algebras, the uniqueness of the related generalized path algebra and quiver holds up to isomorphism in the case
when the ideal is admissible. For an artinian algebra, there are two basic algebras, the first is that associated to the algebra
itself; the second is that associated to the correspondent generalized path algebra. In the final part, it is shown that for
a Gabriel-type artinian algebra, the first basic algebra is a quotient of the second basic algebra. In the end, we give an
example of a skew group algebra in which the relation between the natural quiver and the ordinary quiver is discussed. 相似文献
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Given a covering Γ of a quiver Δ, we show that the quiver algebra K[Γ] of Γ over a field K is a twisted tensor product of the quiver algebra of the fibre of the covering viewed as a trivial quiver and the quiver algebra K[Δ]. To make sense of this, we first extend the theory of twisted tensor products of algebras to include algebras without units. 相似文献
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Christian Gottlieb 《代数通讯》2013,41(12):4687-4691
Abstract Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given. 相似文献
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《代数通讯》2013,41(10):3467-3478
In the first part of this article, we describe the projective representations in the category of representations by modules of a quiver which does not contain any cycles and the quiver A ∞ as a subquiver, that is, the so-called rooted quivers. As a consequence of this, we show when the category of representations by modules of a quiver admits projective covers. In the second part, we develop a technique involving matrix computations for the quiver A ∞, which will allow us to characterize the projective representations of A ∞. This will improve some previous results and make more accurate the statement made in Benson (1991). We think this technique can be applied in many other general situations to provide information about the decomposition of a projective module. 相似文献
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Kiyoshi Igusa 《代数通讯》2020,48(4):1671-1696
AbstractFor modules over an artin algebra, a linear stability condition is given by a “central charge” and a nonlinear stability condition is given by the wall-crossing sequence of a “green path.” Finite Harder-Narasimhan stratifications of the module category, maximal forward hom-orthogonal sequences and maximal green sequences, defined using Fomin-Zelevinsky quiver mutation are shown to be equivalent to finite nonlinear stability conditions when the algebra is hereditary. This is the first of a series of three papers whose purpose is to determine all maximal green sequences of maximal length for quivers of affine type A and determine which are linear. 相似文献
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Julian Külshammer 《Algebras and Representation Theory》2017,20(5):1215-1238
In this paper, we generalise part of the theory of hereditary algebras to the context of pro-species of algebras. Here, a pro-species is a generalisation of Gabriel’s concept of species gluing algebras via projective bimodules along a quiver to obtain a new algebra. This provides a categorical perspective on a recent paper by Geiß et al. (2016). In particular, we construct a corresponding preprojective algebra, and establish a theory of a separated pro-species yielding a stable equivalence between certain functorially finite subcategories. 相似文献
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We develop the fundamentals of hereditary noetherian categories with Serre duality over an arbitrary field k, where the category of coherent sheaves over a smooth projective curve over k serves as the prime example and others are coming from the representation theory of finite dimensional algebras. The proper
way to view such a category is to think of coherent sheaves on a possibly non-commutative smooth projective curve. We define
for each such category notions like function field and Euler characteristic, determine its Auslander-Reiten components and
study stable and semistable bundles for an appropriate notion of degree. We provide a complete classification of hereditary
noetherian categories for the case of positive Euler characteristic by relating these to finite dimensional representations
of (locally bounded) hereditary k-algebras whose underlying valued quiver admits a positive additive function.
Dedicated to Otto Kerner on the occasion of his 60th birthday 相似文献
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The Auslander-Reiten quiver of a finite-dimensional associative algebra encodes information about the indecomposable finite-dimensional representations of and their homomorphisms. A component of the Auslander-Reiten quiver is called preprojective if it does not admit oriented cycles and each of its modules can be shifted into a projective module using the Auslander-Reiten translation. Preprojective components play an important role in the present research on algebras of finite and tame representation type. We present an algorithm which detects all preprojective components of a given algebra.
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In this paper we consider serial rings with T-nilpotent prime radical, factor-rings of which by the prime radical are right Noetherian rings. We prove that the prime quiver
of such a ring is a disconnected union of cycles and chains. In the case when the prime quiver of such a serial ring is a
chain the prime radical is nilpotent. For serial rings with nilpotent prime radical we introduce an analogue of Kupisch series.
Presented by Yu. Drozd
Mathematics Subject Classifications (2000) 16P40, 16G10. 相似文献
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The strong no loop conjecture states that a simple module of finite projective dimension over an artin algebra has no non-zero self-extension. The main result of this paper establishes this well known conjecture for finite dimensional algebras over an algebraically closed field. 相似文献
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HuaLin Huang 《中国科学 数学(英文版)》2012,55(10):2067-2080
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The structure of weak Hopf algebras corresponding to U q (sl 2) are classified by their algebra structure and coalgebra structure. The algebra structure of weak Hopf algebras corresponding to U q (sl 2) can be written as the direct sum of U q (sl 2) and an algebra of polynomials. The coalgebra structure of weak Hopf algebras corresponding to U q (sl 2) are classified by their Ext quiver. There are four types of such structures. 相似文献
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In several branches of representation theory, the existence of Auslander-Reiten sequences has led to new structural insights. for example, in the module theory of artinian algebras [11, 6], in the theory of lattices over classical orders [18, 2] over a complete discrete valuation domain R, and for the corresponding derived categories [12, 17]. For an R-order ? in a finite dimentional algebra A over the quotient field K of R, Auslander and Reiten [2, 5] have charaterized the non-projective indecomposable ?-lattice E for which an Auslander-Reiten sequence (AR-sequence for short) L → H → E exists as those ?-lattices A-module KE is projective. In the present paper, we shall introduce a modified version of AR-sequences in the category ?-lat of ?-lattices which behave similar to AR-sequence of modules over artinian algebras. In fact, there will be a close relationship to AR-sequences in ?-mod, where ? : = ?/(RadR)?. This relationship extends to AR-sequences in A-mod if ? is hereditary (e.g. for a path order ? = R△ of a quiver △ without oriented cycles.) Our investigation is inspired by recent work of W. Crawley-Boevey [9] who determined the lattices E with Extras (E, E = 0 over a path order R△. 相似文献
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V. V. Kirichenko 《Journal of Mathematical Sciences》2005,131(6):6032-6051
The concept of a quiver, i.e., a direct graph of a finite-dimensional algebra, was introduced by Gabriel in connection with
problems of representation theory of such algebras. The notion of a scheme (=quiver Q(A)) of a semiperfect, right Noetherian ring A was introduced by the author and was applied to the study of the structure of serial rings. We give a short review of some
results on structural ring theory, which were obtained by using the graph technique.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra,
2004. 相似文献