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 $T_r(A)$ 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.