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Bases for free Lie superalgebras

Michael Vaughan-Lee

Abstract

We describe a basis for free Lie superalgebras which uses the theory of basic commutators. The only description for bases for free Lie superalgebras that I have found in the literature is in the book "Infinite dimensional Lie superalgebras" by Bahturin et al. Their bases make use of the theory of Shirshov bases in free Lie algebras, and I believe that there is a case for writing up an alternative approach using basic commutators.

Bases for free Lie superalgebras

Abstract

We describe a basis for free Lie superalgebras which uses the theory of basic commutators. The only description for bases for free Lie superalgebras that I have found in the literature is in the book "Infinite dimensional Lie superalgebras" by Bahturin et al. Their bases make use of the theory of Shirshov bases in free Lie algebras, and I believe that there is a case for writing up an alternative approach using basic commutators.
Paper Structure (3 sections, 6 theorems, 39 equations)

This paper contains 3 sections, 6 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Lie superalgebras
  3. Proof of Theorem 2

Key Result

Theorem 1

The free Lie superalgebra $L$ is a free $R$-module with basis $S\cup\{[w,w]\,|\,w\in S$ is odd$\}$.

Theorems & Definitions (6)

  • Theorem 1: Bahturin et al. [1]
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Corollary 6