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Classification of abelian complex structures on 6-dimensional Lie algebras

A. Andrada, M. L. Barberis, I. G. Dotti

Abstract

We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parameterize, on each of these algebras, the space of such structures up to holomorphic isomorphism.

Classification of abelian complex structures on 6-dimensional Lie algebras

Abstract

We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parameterize, on each of these algebras, the space of such structures up to holomorphic isomorphism.

Paper Structure

This paper contains 11 sections, 17 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complex structures on Lie algebras
  4. Abelian complex structures on Lie algebras
  5. Dimension less than or equal to four
  6. Dimension six: the nilpotent case
  7. Equivalence classes of complex structures
  8. Dimension six: the general case
  9. $\dim\, {\mathfrak s}'_J=2$
  10. $\dim\, {\mathfrak s}'_J=4$
  11. ${\mathfrak s} '_J={\mathfrak s}$

Key Result

Proposition 2.1

Figures (1)

  • Figure 1: Orbits in $\mathcal{C}_a({\mathfrak n} _7)$

Theorems & Definitions (39)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5: Sn
  • Proposition 2.6
  • proof
  • ...and 29 more