Point modules over the universal enveloping algebras of color Lie algebras

Abstract

Let $k$ be an algebraically closed field with characteristic zero. In this paper, we define the notion of a $q'$-Heisenberg normal element of a $\mathbb{Z}$-graded $k$-algebra. This $q'$-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.