Kähler Lie algebras and double extension
type: Item
instance of: scholarly article
title: Kähler Lie algebras and double extension
zbMATH Open document ID: 0866.17006
author: Jean-Michel Dardié
author: A. Medina
published in: Journal of Algebra
publication date: 1997-07-27T00:00:00Z
review text: A finite-dimensional real Lie algebra \({\mathfrak g}\) with a nondegenerate 2-cocycle \(\omega\) is called a symplectic Lie algebra. It is called a Kähler-Lie algebra if it also carries an integrable complex structure \(J\) such that \(\omega(X,JY)\) is a scalar product on \({\mathfrak g}\). Kähler-Lie algebras are the infinitesimal analogue of Lie groups with a left invariant Kähler structure. The authors introduce the Kähler double extension process (by means of two exact sequences), which leads to the following structure theorems on Kähler-Lie algebras. Every such algebra \({\mathfrak g}\) can be obtained from a flat Kähler algebra, or from \(\{0\}\), by successive Kähler double extensions. The latter case occurs if and only if \({\mathfrak g}\) admits a Lagrangian subalgebra. Completely solvable Kähler-Lie algebras are obtained from \(\{0\}\) by successive double extensions by means of one-dimensional algebras. They are unimodular (i.e. \(\text{tr(ad }X)\) vanishes identically) if and only if they are abelian.
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 954782
zbMATH Keywords: completely solvable Kähler-Lie algebras
zbMATH Keywords: Kähler double extension process
reviewed by: François Rouvière
Wikidata QID: Q115396066
MaRDI profile type: Publication
full work available at URL: jabr.1996.0350
OpenAlex ID: W2080755014
DOI: 10.1006/JABR.1996.0350
description: scientific article; zbMATH DE number 954782