  This is a general study of twisted Calabi-Yau algebras that are
$\mathbb{N}$-graded and locally finite-dimensional, with the following major
results. We prove that a locally finite graded algebra is twisted Calabi-Yau if
and only if it is separable modulo its graded radical and satisfies one of
several suitable generalizations of the Artin-Schelter regularity property,
adapted from the work of Martinez-Villa as well as Minamoto and Mori. We
characterize twisted Calabi-Yau algebras of dimension 0 as separable
$k$-algebras, and we similarly characterize graded twisted Calabi-Yau algebras
of dimension 1 as tensor algebras of certain invertible bimodules over
separable algebras. Finally, we prove that a graded twisted Calabi-Yau algebra
of dimension 2 is noetherian if and only if it has finite GK dimension.

Rings and Algebras;Quantum Algebra