论文标题
定期琐碎的延伸代数和分数卡拉比远代数
Periodic trivial extension algebras and fractionally Calabi-Yau algebras
论文作者
论文摘要
我们研究了有限维代数$ a $的微不足道扩展代数$ t（a）$的周期性和扭曲的周期性。我们的主要结果表明，$ t（a）$的（扭曲）周期性相当于有限的全球尺寸的$ a $ a $（扭曲的）calabi-yau。我们还将此结果扩展到一大批自注射轨道代数。这是一个重要的结果，这些结果对埃尔德曼·斯科罗斯基的周期性猜想给出了部分答案，后者期望周期性和扭曲的周期性代数类别重合。在实际方面，它使我们能够构建大量的定期代数和分数的新示例。我们还通过表明$ t（a）$的扭曲周期性与$ r $ r $ r $ fold的琐事代数$ t_r（a）$ r，d $ r，d $ r，d $ r，d $ r，d $ r，d \ ge 1 $之间，我们还建立了周期性和集群倾斜理论之间的联系。这回答了Darpö和Iyama的问题。 作为我们结果的应用，我们对其他一些开放问题给出了答案。我们构建了野生代表类型的周期性对称代数，其任意最小的时期，回答了Skowroński的问题。我们还表明，在派生的等效性下，扭曲的扭曲的calabi-yau代数是封闭的，回答了赫尔施德和伊玛的一个问题。
We study periodicity and twisted periodicity of the trivial extension algebra $T(A)$ of a finite-dimensional algebra $A$. Our main results show that (twisted) periodicity of $T(A)$ is equivalent to $A$ being (twisted) fractionally Calabi-Yau of finite global dimension. We also extend this result to a large class of self-injective orbit algebras. As a significant consequence, these results give a partial answer to the periodicity conjecture of Erdmann-Skowroński, which expects the classes of periodic and twisted periodic algebras to coincide. On the practical side, it allows us to construct a large number of new examples of periodic algebras and fractionally Calabi-Yau algebras. We also establish a connection between periodicity and cluster tilting theory, by showing that twisted periodicity of $T(A)$ is equivalent the $d$-representation-finiteness of the $r$-fold trivial extension algebra $T_r(A)$ for some $r,d\ge 1$. This answers a question by Darpö and Iyama. As applications of our results, we give answers to some other open questions. We construct periodic symmetric algebras of wild representation type with arbitrary large minimal period, answering a question by Skowroński. We also show that the class of twisted fractionally Calabi-Yau algebras is closed under derived equivalence, answering a question by Herschend and Iyama.
