自内射Nakayama代数的q-Cartan矩阵

doi:10.6040/j.issn.1671-9352.0.2020.256

《山东大学学报(理学版)》 ›› 2020, Vol. 55 ›› Issue (10): 46-51.doi: 10.6040/j.issn.1671-9352.0.2020.256

自内射Nakayama代数的q-Cartan矩阵

赵跳,章超^*

贵州大学数学与统计学院, 贵州贵阳 550025

出版日期:2020-10-20 发布日期:2020-10-07
作者简介:赵跳(1995— ), 男, 硕士研究生, 研究方向为代数表示论. E-mail:1475759283@qq.com*通信作者简介:章超(1986— ), 男,博士, 教授, 研究方向为代数表示理论. E-mail:zhangc@amss.ac.cn
基金资助:
国家自然科学基金资助项目(11961007);贵州省科技厅项目(黔科合基础［2020］1Y405)

q-Cartan matrices of self-injective Nakayama algebras

ZHAO Tiao, ZHANG Chao^*

School of Mathematics and Statistics, Guizhou University, Guiyang 550025, Guizhou, China

Online:2020-10-20 Published:2020-10-07

摘要/Abstract

摘要： 证明了任意自内射的Nakayama代数A的q-Cartan矩阵C_A(q)相似于一个对角矩阵,而且C_A(q)的行列式为|C_A(q)|={(1-(qⁿ)^m)/(1-qⁿ),〓〓〓如果(n,m)=1;((1-q^［m,n］)^(m,n))/(1-qⁿ),如果(n,m)≠1,其中n为单模的个数,m为齐次关系理想I中最短路径的长度,(n,m)表示n与m的最大公因数,［n,m］表示n与m的最小公倍数。

关键词: Cartan行列式, 可对角化矩阵, 循环矩阵

Abstract: The present paper mainly proves that the q-Cartan matrix of any self-injective Nakayama algebra A is diagonalizable and the determinant of q-Cartan|C_A(q)|={(1-(qⁿ)^m)/(1-qⁿ), if(n,m)=1;((1-q^［m,n］)^(m,n))/(1-qⁿ),if (n,m)≠1,where n is the number of simple modules, m is the length of the shortest paths in the homogeneous ideal I, and (n,m) is the greatest common divisor of n and m, ［n,m］ is the least common multiple of n and m.

Key words: Cartan determinant, diagonalizable matrix, circulant matrix

中图分类号:

O153.3

赵跳,章超. 自内射Nakayama代数的q-Cartan矩阵[J]. 《山东大学学报(理学版)》, 2020, 55(10): 46-51.

ZHAO Tiao, ZHANG Chao. q-Cartan matrices of self-injective Nakayama algebras[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(10): 46-51.

参考文献

[1] EILENBERG S. Algebras of cohomologically finite dimension[J]. Commentarii Mathematici Helvetici, 1954, 28(1):310-319.
[2] JANS J P, NAKAYAMA T. On the dimension of modules and Algebras, VII. Algebras with finite-dimensional residue-algebras[J]. Nagoya Mathematical Journal, 1957, 11:67-76.
[3] ZACHARIA D. On the Cartan matrix of an Artin algebra of global dimension two[J]. Journal of Algebra, 1983, 82(2):353-357.
[4] BURGESS W D, FULLEER K R, VOSS E R, et al. The Cartan matrix as an indicator of finite global dimension for Artinian rings[J]. Proceedings of the American Mathematical Society, 1985, 95(2):157-165.
[5] HOLM T. Cartan determinants for gentle algebras[J]. Archiv der Mathematik, 2005, 85(3):233-239.
[6] BESSENRODT C, HOLM T. q-Cartan matrices and combinatorial invariants of derived categories for skewed-gentle algebras[J]. Pacific Journal of Mathematics, 2007, 229(1):25-47.
[7] ASSEM I, SIMSON D, SKOWRONSKI A. Elements of the representation theory of associative algebras: Vol.1: techniques of representation theory[M]. Cambridge: Cambridge University Press, 2006.
[8] GUSTAFSON W H. Global dimension in serial rings[J]. Journal of Algebra, 1985, 97(1):14-16.
[9] MADSEN D. Projective dimensions and Nakayama algebras[J]. Representations of Algebras and Related Topics, 2005: 247-265.
[10] RINGEL C M. The Gorenstein projective modules for the Nakayama algebras. I[J]. Journal of Algebra, 2013, 385:241-261.
[11] CHEN Xiao-wu, YE Yu. Retractions and Gorenstein homological properties[J]. Algebras and Representation Theory, 2014, 17(3):713-733.
[12] SHEN Dawei. The singularity category of a Nakayama algebra[J]. Journal of Algebra, 2015, 429:1-18.
[13] 谢宗真, 张孝金, 昝立博. 自内射的Nakayama代数的嘉当矩阵[J]. 南京大学学报(数学半年刊), 2017, 34(2):155-169. XIE Zongzhen, ZHANG Xiaojin, ZAN Libo. The Cartan matrice of self-injective Nakayama algebras[J]. Journal of Nanjing University(Mathematical Biquarterly), 2017, 34(2):155-169.

多维度评价

Viewed

Full text

Abstract

Cited

Shared

Discussed

本文评价

推荐阅读 10

[1]	张京友,张培爱,钟海萍. 进化图论在知识型企业组织结构设计中的应用[J]. J4, 2013, 48(1): 107 -110 .
[2]	史开泉. 信息规律智能融合与软信息图像智能生成[J]. 山东大学学报（理学版）, 2014, 49(04): 1 -17 .
[3]	章玲,周德群 . λ模糊测度及其Mbius变换和关联系数间关系的推导[J]. J4, 2007, 42(7): 33 -37 .
[4]	王碧玉,曹小红*. 算子矩阵的Browder定理的摄动[J]. 山东大学学报（理学版）, 2014, 49(03): 90 -95 .
[5]	胡选子1, 谢存禧2. 基于人工免疫网络的机器人局部路径规划[J]. J4, 2010, 45(7): 122 -126 .
[6]	付海艳,卢昌荆,史开泉 . (F,F-)-规律推理与规律挖掘[J]. J4, 2007, 42(7): 54 -57 .
[7]	. Jordan代数上的三元映射[J]. J4, 2009, 44(6): 1 -3 .
[8]	闫鹏,沈艳军,王建凤. 一类广义中立型系统的稳定性分析[J]. J4, 2009, 44(4): 66 -71 .
[9]	安云鹤, 蔡召平, 张伟, 黄淑红, 杨凌, 张红卫. 文昌鱼Rbx1同源基因的克隆，进化分析和表达图式的研究（英文）[J]. J4, 2009, 44(3): 11 -16 .
[10]	房用,慕宗昭,王月海,鲍玉海,孙蕾 . 个杨树无性系蒸腾特性及其影响因子研究[J]. J4, 2006, 41(6): 168 -172 .

自内射Nakayama代数的q-Cartan矩阵

q-Cartan matrices of self-injective Nakayama algebras

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 2

多维度评价

本文评价

推荐阅读 10

[1]	刘静姝,王莉,刘惊雷. 基于循环矩阵投影的Nyström扩展[J]. 《山东大学学报(理学版)》, 2020, 55(7): 55-66.
[2]	李静,何承源. 求解首尾差循环矩阵逆与广义逆的快速算法[J]. J4, 2013, 48(6): 96-99.