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A computational approach to almost-inner derivations

Heiko Dietrich, Willem A. de Graaf

TL;DR

A computational approach is presented to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table and an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian.

Abstract

We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian. This answers a question of Kunyavskii and Ostapenko.

A computational approach to almost-inner derivations

TL;DR

A computational approach is presented to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table and an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian.

Abstract

We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian. This answers a question of Kunyavskii and Ostapenko.
Paper Structure (4 sections, 3 theorems, 24 equations, 2 figures)

This paper contains 4 sections, 3 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Computational approach
  3. Computational examples
  4. A non-abelian $\Sha(\mathfrak{g})$

Key Result

Proposition 2.2

The derivation $\delta\in V$ lies in ${\rm AID}(\mathfrak{g})$ if and only if for all $z\in F^n$

Figures (2)

  • Figure 1: Structure constants for $\mathfrak{g}_3$; if $[v_i,v_j]$ is not listed, then $[v_i,v_j]=0$.
  • Figure 2: Definitions of two non-commuting elements of $U$; if $d_i(v_k)$ is not listed, then $d_i(v_k)=0$.

Theorems & Definitions (10)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Remark 2.5
  • Example 3.1
  • Example 3.2
  • Example 3.3