Untwisting a twisted Calabi-Yau algebra

TL;DR

This paper demonstrates that twisted Calabi-Yau algebras can be extended to Calabi-Yau algebras through specific smash product constructions, broadening the understanding of their structure.

Contribution

It proves that twisted Calabi-Yau algebras can be transformed into Calabi-Yau algebras via smash product extensions with automorphisms.

Findings

A twisted Calabi-Yau algebra with automorphism σ can be extended to a Calabi-Yau algebra.

The smash product algebras A ⋉_σ N and A ⋉_σ Z are Calabi-Yau.

This extension provides a method to 'untwist' twisted Calabi-Yau algebras.

Abstract

Twisted Calabi-Yau algebras are a generalisation of Ginzburg's notion of Calabi-Yau algebras. Such algebras A come equipped with a modular automorphism \sigma \in Aut(A), the case \sigma = id being precisely the original class of Calabi-Yau algebras. Here we prove 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 \sigma, then the smash product algebras A \rtimes_\sigma N and A \rtimes_\sigma Z are Calabi-Yau.