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Historically, the study of graded (twisted or otherwise) Calabi-Yau algebras has meant the study of such algebras under an 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 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.
Yasmeen S. Baki (Tue,) studied this question.