北京大学期刊网　|　作者　　审稿人　　编委专家　　工作人员 首页　  |  　关于　  |  　浏览　  |  　投稿指南　  |  　新闻公告
 数学进展
 研究论文
 扭平凡扩张与表示维数 Twisted Trivial Extension and Representation Dimension of the Complete Bipartite Graph mbox{boldmath $K_{m,n}$}} 郑立景 ZHENG Lijing 1. 湖南师范大学数学与计算机科学学院, 长沙, 湖南, 410081; 2. 赣南师范学院数学与计算机科学学院, 赣州, 江西, 341000 1. College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan, 410081, P. R. China; 2.School of Mathematics and Computer Science, Gannan Normal University, Ganzhou, Jiangxi, 341000, P. R. China 收稿日期: 2013-06-28 出版日期: 2014-07-25 DOI: 10.11845/sxjz.2013081b
 168 浏览 引用导出
0
/   /   推荐
 摘要 设$k$是代数闭域,$Lambda$是$k$上基本有限维连通Koszul自入射代数.本文首先证明:如果$Lambda$满足有限生成(FG)假设,那么存在$Lambda$的$k$-代数自同构$sigma_{0}$使得关于$Lambda$-双模$DLambda^{sigma_{0}}$的扭平凡扩张$T(Lambda^{sigma_{0}})=LambdaltimesDLambda^{sigma_{0}}$亦满足FG假设.由此得到,在$Lambda$满足FG假设的条件下,(1)$T(Lambda^{sigma_{0}})$的表示维数大于等于$Lambda$的复杂度加2;(2)设$G$是$Lambda$的$k$-代数自同构群Aut$_{k}(Lambda)$的有限子群,且其阶在$Lambda$中可逆.如果对于任意的$ginG$都有$sigma_{0}g=gsigma_{0}$,那么斜群代数$LambdaastG$的扭平凡扩张代数$T((LambdaastG)^{widetilde{sigma_{0}}})$的表示维数大于等于$Lambda$的复杂度加2. 关键词 ： 表示维数,  有限生成假设,  扭平凡扩张,  Koszul自入射代数 Abstract：Let $k$ be an algebraically closed field and $Lambda$ be a finite dimensional basic connected Koszul selfinjective algebra. In this paper, we first prove that if Lambda$satisfies FG assumption, then there exists a$k$-algebra automorphism$sigma_{0}$such that$T(Lambda^{sigma_{0}})=Lambdaltimes DLambda^{sigma_{0}}$, the trivial extension of$Lambda$-bimodule$DLambda^{sigma_{0}}=D(_{Lambda}Lambda^{sigma_{0}}_{Lambda})$,also satisfies FG assumption. Under the condition that$Lambda$satisfies FG assumption, we obtain the following results: (1) The representation dimension of$T(Lambda^{sigma_{0}})$is not less than the complexity of$Lambda$plus 2; (2) Let$G$be a finite subgroup of Aut$_{k}(Lambda)$, and its order be invertible in$Lambda$. If$gsigma_{0}=sigma_{0}g$for any$gin G$, then the representation dimension of$T((Lambdaast G)^{widetilde{sigma_{0}}})$is not less than the complexity of$Lambda\$ plus 2. Key words： FG assumption    twisted trivial extension    Koszul selfinjective algebra 基金资助:国家自然科学基金(No.11271119)和湖南省研究生科研创新项目(No.CX2012B199).
 [1] 惠昌常;. 关于有限维数猜想的一些新进展[J]. 数学进展, 2007, 36(1): 13-17.
Viewed
Full text

Abstract

Cited

Discussed