关于n-平移代数和n-阿贝尔范畴的研究
发布时间：2019-07-02 20:50
【摘要】：代数表示论高维理论是Iyama等人推广经典Auslander-Reiten理论,引入n-Auslander代数[63], n-Auslander-Reiten平移函子[64]等建立发展起来的.作为平移代数的推广,郭引入了n-平移代数,并揭示了n-平移代数与高维表示理论的内在联系[56].本文主要讨论两个方面的问题：一是研究(n+1)-表示有限的n-Auslander代数(更一般的,n-预平移立方代数)及其Koszul对偶的平移性、周期性、几乎Koszul性和n-APR倾斜性等性质；二是利用n-阿贝尔范畴刻画n-Auslander代数.具体组织如下：我们首先研究n-平移代数,第三章我们研究n-立方代数,计算它们与扭平凡扩张代数的单模投射分解.第四章我们从(n+1)-表示有限的n-Auslander代数出发引入(稳定)n-金字塔代数,并研究它们及其Kosuzl对偶的Koszul性、周期性、平移性.第五章我们研究Υ[n]n-mutation,对于整体维数小于等于n的Koszul代数,如果其Kosuzl对偶为允许(n-1)-平移代数,则其n-APR倾斜为原代数的T[n]-mutation.在第六章,我们研究高维Auslander对应.作为Iyama-Beligiannis的n-Auslander对应的推广,我们利用n-阿贝尔范畴给出n-Auslander 对应,从而给出一个利用n-阿贝尔范畴的n-Auslander范畴的刻画.并在此基础上证明一个加法范畴为具有拟足够多内射对象的n-阿贝尔范畴当且仅当它能够作为n-丛倾斜子范畴嵌入到内射上生成的阿贝尔范畴中.
[Abstract]:The higher dimensional theory of algebra representation theory was developed by Iyama et al., which extended the classical Auslander-Reiten theory and introduced n-Auslander algebra [63] and n-Auslander-Reiten translation functor [64]. As a generalization of translation algebra, Guo introduced n-translation algebra and revealed the internal relationship between n-translation algebra and high-dimensional representation theory [56]. In this paper, we mainly discuss two problems: one is to study (n 1)-denoted finite n-Auslander algebra (more general, n-pretranslation cubic algebra) and its Koszul duality translation, periodicity, almost Koszulicity and n-APR inclination, and the other is to characterize n-Auslander algebra by using n-Abel category. The specific organization is as follows: we first study n-translation algebra, in chapter 3, we study n-cubic algebra, and calculate the single-mode projective decomposition of n-cubic algebra and torsional trivial extension algebra. In chapter 4, we introduce (stable) n-pyramid algebra from (n 1)-finite n-Auslander algebra, and study the Koszul property, periodicity and translation property of them and their Kosuzl duality. In chapter 5, we study the mutation of r [n] n. For Koszul algebra whose global dimension is less than or equal to n, if its Kosuzl duality is allowable (n 鈮，
本文编号：2509224
[Abstract]:The higher dimensional theory of algebra representation theory was developed by Iyama et al., which extended the classical Auslander-Reiten theory and introduced n-Auslander algebra [63] and n-Auslander-Reiten translation functor [64]. As a generalization of translation algebra, Guo introduced n-translation algebra and revealed the internal relationship between n-translation algebra and high-dimensional representation theory [56]. In this paper, we mainly discuss two problems: one is to study (n 1)-denoted finite n-Auslander algebra (more general, n-pretranslation cubic algebra) and its Koszul duality translation, periodicity, almost Koszulicity and n-APR inclination, and the other is to characterize n-Auslander algebra by using n-Abel category. The specific organization is as follows: we first study n-translation algebra, in chapter 3, we study n-cubic algebra, and calculate the single-mode projective decomposition of n-cubic algebra and torsional trivial extension algebra. In chapter 4, we introduce (stable) n-pyramid algebra from (n 1)-finite n-Auslander algebra, and study the Koszul property, periodicity and translation property of them and their Kosuzl duality. In chapter 5, we study the mutation of r [n] n. For Koszul algebra whose global dimension is less than or equal to n, if its Kosuzl duality is allowable (n 鈮，
本文编号：2509224
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