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Point modules over the universal enveloping algebras of color Lie algebras and overlapping normal elements

Shu Minaki

Abstract

Let $k$ be an algebraically closed field with characteristic $0$. In this paper, we define the notion of an overlapping normal element of a $\mathbb{Z}$-graded $k$-algebra. This overlapping normal element gives us the two criteria for the nonexistence of a module. The first one involves a module with low Gelfand--Kirillov dimension. Also, the second one involves a truncated point module. By using the second one, we determine the set of point modules over an Artin--Schelter regular algebra obtained as the universal enveloping algebra of a color Lie algebra.

Point modules over the universal enveloping algebras of color Lie algebras and overlapping normal elements

Abstract

Let be an algebraically closed field with characteristic . In this paper, we define the notion of an overlapping normal element of a -graded -algebra. This overlapping normal element gives us the two criteria for the nonexistence of a module. The first one involves a module with low Gelfand--Kirillov dimension. Also, the second one involves a truncated point module. By using the second one, we determine the set of point modules over an Artin--Schelter regular algebra obtained as the universal enveloping algebra of a color Lie algebra.

Paper Structure

This paper contains 9 sections, 16 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graded algebras and filtered algebras
  4. Point schemes and point modules
  5. Color Lie algebras
  6. Overlapping normal elements
  7. Overlapping normal elements and point modules
  8. Overlapping normal elements and truncated point modules
  9. Point modules over the universal enveloping algebras of color Lie algebras

Key Result

Lemma 2.2

For a positive integer $r \geq 1$, let $A=\bigoplus_{i \in \mathbb{Z}}A_{i}$ be a $\mathbb{Z}$-graded $k$-algebra and $A^{[r]}$ the $r$-th quasi-Veronese algebra of $A$. Then the following categorical equivalence holds: This categorical equivalence is given by the functor $Q: A\mathchar'-\mathsf{GrMod}\, \rightarrow A^{[r]}\mathchar'-\mathsf{GrMod}\,$; with the action $(a_{i,j})(m_{i})=\left(\math

Theorems & Definitions (40)

  • Definition : Definition \ref{['dfn-one']}
  • Definition 2.1: Mori
  • Lemma 2.2: Mori
  • Definition 2.3: Zhang
  • Definition 2.4: Zhang
  • Proposition 2.5: Zhang
  • Definition 2.6: see HvO
  • Lemma 2.7: see HvO
  • Proposition 2.8: See HvO
  • Definition 2.9: ATV90OF
  • ...and 30 more