Periodic trivial extension algebras and fractionally Calabi-Yau algebras
Periodic trivial extension algebras and fractionally Calabi-Yau algebras
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发表时间：
2020-12
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通讯作者：
Aaron Chan;Erik Darpo;O. Iyama;René Marczinzik
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作者：
Aaron Chan;Erik Darpo;O. Iyama;René Marczinzik
研究了有限维代数A的平凡扩展代数T(A)$的周期性和扭曲周期性。我们的主要结果表明，$T(A)$的（扭曲）周期性等价于$A$是有限整体维数的（扭曲）部分Calabi-Yau。我们也将这一结果推广到一大类自注入轨道代数。作为一个重要的结果，这些结果给出了Erdmann-Skowro\'nski的周期性猜想的部分答案，该猜想期望周期代数和扭曲周期代数的类重合。在实际应用方面，它允许我们构造大量周期代数和分数型Calabi-Yau代数的新例子。我们还通过证明$T(a)$的扭曲周期性等价于$r$-倍平凡扩展代数$T_r(a)$的$d$-表示有限性，建立了周期性与簇倾斜理论之间的联系。这回答了Darp\ o和Iyama提出的问题。作为我们的结果的应用，我们给出了一些其他开放性问题的答案。构造了具有任意大最小周期的野表示型周期对称代数，回答了Skowro\'nski的问题。我们还证明了一类扭曲分数型Calabi-Yau代数在派生等价下是闭的，回答了Herschend和Iyama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-Skowro\'nski, 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 Darp\"o 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 Skowro\'nski. We also show that the class of twisted fractionally Calabi-Yau algebras is closed under derived equivalence, answering a question by Herschend and Iyama.