Periodic trivial extension algebras and fractionally Calabi-Yau algebras
Aaron Chan, Erik Darp\"o, Osamu Iyama, Ren\'e Marczinzik

TL;DR
This paper explores the relationship between periodicity and fractional Calabi-Yau properties of trivial extension algebras, establishing new links with cluster tilting theory and providing numerous examples and answers to open questions.
Contribution
It proves that (twisted) periodicity of trivial extension algebras is equivalent to being (twisted) fractionally Calabi-Yau, extending to self-injective orbit algebras and connecting to cluster tilting.
Findings
Periodic and twisted periodic trivial extension algebras are characterized by fractional Calabi-Yau properties.
Constructed new examples of periodic symmetric algebras of wild type with large periods.
Showed that twisted fractionally Calabi-Yau algebras are closed under derived equivalence.
Abstract
We study periodicity and twisted periodicity of the trivial extension algebra of a finite-dimensional algebra . Our main results show that (twisted) periodicity of is equivalent to 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 is equivalent the -representation-finiteness of the -fold trivial extension…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Nonlinear Waves and Solitons
