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Structure Constant Formulas for the Universal Enveloping Algebras of the Nilpotent Lie Algebras of Dimension Five and Less

Samuel Chamberlin, Emmerson Taylor

Abstract

Libor Šnobl and Pavel Winternitz classified all of the Lie algebras of dimension six and smaller. Using this classification, we formulated and proved structure constant formulas for the universal enveloping algebras of the nilpotent Lie algebras of dimension five and less.

Structure Constant Formulas for the Universal Enveloping Algebras of the Nilpotent Lie Algebras of Dimension Five and Less

Abstract

Libor Šnobl and Pavel Winternitz classified all of the Lie algebras of dimension six and smaller. Using this classification, we formulated and proved structure constant formulas for the universal enveloping algebras of the nilpotent Lie algebras of dimension five and less.

Paper Structure

This paper contains 8 sections, 15 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lie Algebras
  4. Nilpotency
  5. The Universal Enveloping Algebra
  6. Straightening Process
  7. Necessary Straightening Identities
  8. Structure Constant Formulas

Key Result

Theorem 2.5

A finite dimensional Lie algebra $L$ is nilpotent if for all $x,y\in L$. In other words, repeated bracketing of $x$ on $y$ eventually gives 0.

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Engel's Theorem
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8: The Poincaré-Birkhoff-Witt (PBW) Theorem
  • Definition 2.9
  • Theorem 2.10: Chamberlin, Peck, and Rafizadeh Theorem
  • ...and 32 more