一个Cluster-Tilted代数的Hochschild上同调环

徐运阁, 赵体伟, 吴迪

数学学报 ›› 2016, Vol. 59 ›› Issue (4) : 505-518.

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PDF(497 KB)
数学学报 ›› 2016, Vol. 59 ›› Issue (4) : 505-518. DOI: 10.12386/A2016sxxb0047
论文

一个Cluster-Tilted代数的Hochschild上同调环

    徐运阁1, 赵体伟2, 吴迪1
作者信息 +

Hochschild Cohomology Ring of a Cluster-Tilted Algebra

    Yun Ge XU1, Ti Wei ZHAO2, Di WU1
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摘要

基于Furuya构造的一个cluster-tilted代数的极小投射双模分解,定义了该投射分解的所谓"余乘"结构,从而证明了该代数的Hochschild上同调环的cup积本质上是平行路的毗连并由此得到了该代数的Hochschild上同调环的一个由生成元与关系给出的实现.

Abstract

In this paper, based on the minimal projective bimodule resolution of a cluster-tilted algebra given by Furuya, we define the so-called "comultiplication" structure of the minimal projective bimodule resolution, and show that the cup product of Hochschild cohomology ring of the cluster-tilted algebra is essentially juxtaposition of parallel paths up to sign. As a consequence, we determine the structure of the Hochschild cohomology ring under the cup product by giving an explicit presentation via generators and relations.

关键词

cluster-tilted代数 / cup积 / Hochschild上同调环 / 平行路

Key words

cluster-tilted algebra / cup product / Hochschild cohomology ring / parallel path

引用本文

导出引用
徐运阁, 赵体伟, 吴迪. 一个Cluster-Tilted代数的Hochschild上同调环. 数学学报, 2016, 59(4): 505-518 https://doi.org/10.12386/A2016sxxb0047
Yun Ge XU, Ti Wei ZHAO, Di WU. Hochschild Cohomology Ring of a Cluster-Tilted Algebra. Acta Mathematica Sinica, Chinese Series, 2016, 59(4): 505-518 https://doi.org/10.12386/A2016sxxb0047

参考文献

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[2] Buchweitz R. O., Green E. L., Snashall N., et al., Multiplicative structures for Koszul algebras, Quart. J. Math., 2008, 59: 441-454.
[3] Cibils C., Rigidity of truncated quiver algebras, Adv. Math., 1990, 79: 18-42.
[4] Furuya T., A projective bimodule resolution and the Hochschild cohomology for a cluster-tilted algebra of type D4, SUT J. Math., 2012, 48: 145-169.
[5] Mac Lane S., Homology, Springer-Verlag, Berlin, 1963.
[6] Siegel S. F., Witherspoon S. J., The Hochschild cohomology ring of a group algebra, Proc. London Math. Soc., 1999, 79: 131-157.
[7] Xu Y. G., Xiang H. L., Hochschild cohomology rings of d-Koszul algebras, J. Pure Appl. Algebra, 2011, 215: 1-12.

基金

国家自然科学基金资助项目(11371186,11571341)

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