Untwisting a twisted Calabi–Yau algebra

Twisted Calabi–Yau algebras are a generalisation of Ginzburg's notion of Calabi–Yau algebras. Such algebras A come equipped with a modular automorphism  σ∈Aut(A)σ∈Aut(A), the case σ=idσ=id being precisely the original class of Calabi–Yau algebras. The aim of this paper is to give a concise proof of the fact that every twisted Calabi–Yau algebra may be extended to a Calabi–Yau algebra. More precisely, we show that if A is a twisted Calabi–Yau algebra with modular automorphism σ  , then the smash (semidirect) product algebras A⋊σNA⋊σN and A⋊σZA⋊σZ are Calabi–Yau.