Group-Graded Twisted Calabi--Yau Algebras

Historically, the study of graded (twisted or otherwise) Calabi--Yau algebras has meant the study of such algebras under an $\mathbb{N}$-grading. In this paper, we propose a suitable definition for a twisted $G$-graded Calabi-Yau algebra, for $G$ an arbitrary abelian group. Building on the work of Reyes and Rogalski, we show that a $G$-graded algebra is twisted Calabi-Yau if and only if it is $G$-graded twisted Calabi--Yau. In the second half of the paper, we prove that localizations of twisted Calabi--Yau algebras at elements which form both left and right denominator sets remain twisted Calabi--Yau. As such, we obtain a large class of $\mathbb{Z}$-graded twisted Calabi--Yau algebras arising as localizations of Artin-Schelter regular algebras. Throughout the paper, we survey a number of concrete examples of $G$-graded twisted Calabi--Yau algebras, including the Weyl algebras, families of generalized Weyl algebras, and universal enveloping algebras of finite dimensional Lie algebras.