一个Cluster-Tilted代数的Hochschild上同调环
|
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.
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.
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.
引文
[1]Bastian J.,Holm T.,Ladkani S.,Towards derived equivalence classification of the cluster-tilted algebras of Dynkin type D,J.Algebra,2014,410:277-332.
[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 D_4,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.
[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 D_4,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.