Point modules over the universal enveloping algebras of color Lie algebras

Let $k$ be an algebraically closed field with characteristic zero. In this paper, we define the notion of a $q^{\mathrm{\prime }}$-Heisenberg normal element of a $\mathbb{Z}$-graded $k$-algebra. This $q^{\mathrm{\prime }}$-Heisenberg normal element gives the structure of some sets of modules related to point modules. We also determine the set of point modules over an Artin–Schelter regular algebra obtained as the universal enveloping algebra of a color Lie algebra. Moreover, we give a concrete integer such that the inverse system of its truncated point schemes is constant. This is a quantitative answer to a question raised by Artin–Tate–Van den Bergh, in our setting.