【作者】 王军东；

【导师】 李思泽；

【作者基本信息】 北京交通大学 ， 基础数学， 2007， 硕士

【摘要】 代数表示论是上世纪70年代初兴起的代数学的一个新的分支，它的基本内容是研究环与代数的结构。在近三十年的时间里这一理论有了异常迅猛的发展并逐步趋于完善。它主要研究一个给定的Artin代数是有限型还是无限型，若是无限型，给出模的分布情况，若是有限型，确定其全体不可分解模，通过研究不可分解模之间的关系并结合A-R箭图，对代数进行分类。 近几年，尤其是对有限维代数的研究取得了一系列较好的结果。1982年，D．Happel和C．Ringel在遗传代数(hereditary algebras)的基础上定义了倾斜代数(tilted algebras)，使倾斜代数成为代数表示论中非常重要的一种代数类型，而通过对已知代数类型进行扩展研究以发现更一般的代数类型也成为了一种很有效的方法。1996年，倾斜代数被一般化为拟倾斜代数。拟倾斜代数有许多良好的性质，我们感兴趣的是：Artin代数∧是拟倾斜代数当且仅当任何一条从不可分解内射∧-模到不可分解投射∧-模的路(简称IP路)能被加细成一条既约映射路，且这条路是section路。1999年，拟倾斜代数更一般化为shod代数[7]。和拟倾斜代数类似，Artin代数∧是shod代数当且仅当任何一条IP路能被加细成一条既约映射路，且任何这样的加细至多有两个钩子，如果有两个的话，它们是连续的。 由文献[7]，shod代数的赋值箭图无有向循环，因此它在本文的研究范围之内。所谓严格shod代数，就是整体维数等于3的shod代数，因此严格shod代数除了具有shod代数的一切特性外，还应具有某些特殊的性质。第三章定理3.3.2证明了严格shod代数中一定存在这样的既约映射IP路：使得P(?)L_∧，且这样的路至少带有一个钩子，但至多带有两个钩子，这两个钩子(如果存在的话)是连续的。本文第四章讨论不包含有向循环的A-R分支的嵌入，先讨论正则分支与(非)半正则分支、hip-bounded分支等，然后利用李定理[12]，给出几种满足一定条件的A-R分支的嵌入，同时还给出截面(section)的一类性质[4.3.10]：令∧是一个Artin代数，Γ是Γ_∧的分支。若Γ包含一个截面△，则Γ＼△(?)L_∧ ∪R_∧。更多还原

【Abstract】 Represetation theory of algebra is a new branch of algebra started in early 70’s in last century. Its basic content is to study the structure of rings and algebras.In the last thirty years, this theory has got a great development and maturates gradually. Its main research is to study whether a given Artin algebra is represetation-finite or not. If it is infinite, we show the distribution of modules; If it is finite, find its all indecomposable modules and classify algebras according to the relationship between indecomposable modules and AR-quiver.Over the years, a series of results has been gotten, especially in the study of finite-dimension algebras. In 1982, D.Happel and C.Ringel introduced tilted algebras as a generalization of hereditary algebras. The tilted algebras is now considered to be one of the most useful algebras in represetation theory of algebras. In order to find more general class of algebras,extending and researching the preceding class is an effective method. In 1996, the quasi-tilted algebras generalized the tilted algebras. Quasi-tilted algebras have several interesting characterizations. The most relevant one for this paper is that A is quasi-tilted if and only if any path from an indecomposable injective A-module to an indecomposable projective A-module can be refined to a path of irreducible maps, and any such refinement is sectional. As a generalization of the quasi-tilted algebras, the class of shod algebras was introduced in 1999. Similarly as quasi-tilted algebras, an Artin algebra A is a shod algebra if and only if any path from an indecomposable injective module to an indecomposable projective module can be refined to a path of irreducible maps, and any such refinement has at most two hooks and if there are two, then they must be consecutive.In [7], we know that the valued quiver of shod algebras has no oriented cycles, so it is within our research. The strict shod algebra is the shod algebra whose global dimension is 3, therefore, except for having all the characterizations of shod algebra, the strict shod algebra should have some special properties. In chapter 3 of this paper, we have proved that there must be exist such irreducible IP path in strict shod algebras: the path such that P(?)L_Λ, and it has at least one hook, but has at most two hooks, moreover, if there are two, they are consecutive. In更多还原