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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

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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.
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