# Periodic trivial extension algebras and fractionally Calabi-Yau algebras

We study periodicity and twisted periodicity of the trivial extension algebra T(A) of a finite-dimensional algebra A. We prove that (twisted) periodicity of the trivial extension is equivalent to A being (twisted) fractionally Calabi–Yau. Moreover, twisted periodicity of T(A) is equivalent to the d-representation-finiteness of the r-fold trivial extension algebra Tr(A) for some positive integers r and d. These results allow us to construct a large number of new examples of periodic as well as fractionally Calabi–Yau algebras, and give answers to several open questions. This is a joint work with Aaron Chan, Erik Darpö and René Marczinzik.