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An approach to annihilators in the context of vector field Lie algebras

Charles H. Conley, William Goode

Abstract

We present a general method for describing the annihilators of modules of Lie algebras under certain conditions, which hold for some tensor modules of vector field Lie algebras. As an example, we apply the method to obtain an efficient proof of previously known results on the annihilators of the bounded irreducible modules of Vec(R).

An approach to annihilators in the context of vector field Lie algebras

Abstract

We present a general method for describing the annihilators of modules of Lie algebras under certain conditions, which hold for some tensor modules of vector field Lie algebras. As an example, we apply the method to obtain an efficient proof of previously known results on the annihilators of the bounded irreducible modules of Vec(R).
Paper Structure (12 sections, 30 theorems, 81 equations)

This paper contains 12 sections, 30 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Annihilators generated in a single degree
  4. $\mathrm{Vec} \space \mathbb R$ and the tensor density modules
  5. The structure of ${\mathfrak U}(\mathrm{Vec} \space \mathbb R)$
  6. The structure of ${\mathfrak U}_2(\mathrm{Vec} \space \mathbb R)$
  7. Annihilators over $\mathrm{Vec} \space \mathbb R$ generated in degree 2
  8. Annihilators generated in multiple degrees
  9. The structure of ${\mathfrak U}_3(\mathrm{Vec} \space \mathbb R)$
  10. Annihilators over $\mathrm{Vec} \space \mathbb R$ generated in degrees 2 and 3
  11. Remarks and questions
  12. Related work

Key Result

Proposition 3.1

Let $\pi$ be a representation of ${\mathfrak g}$ on a space $V$ such that there exist satisfying the following conditions: Then

Theorems & Definitions (65)

  • Definition
  • Definition
  • Definition
  • Definition
  • Remark
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Definition
  • ...and 55 more