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Full Description
A co-publication of the AMS and Centre de Recherches Mathématiques.
The purpose of this book is to serve as a tool for researchers and practitioners who apply Lie algebras and Lie groups to solve problems arising in science and engineering. The authors address the problem of expressing a Lie algebra obtained in some arbitrary basis in a more suitable basis in which all essential features of the Lie algebra are directly visible. This includes algorithms accomplishing decomposition into a direct sum, identification of the radical and the Levi decomposition, and the computation of the nilradical and of the Casimir invariants. Examples are given for each algorithm.
For low-dimensional Lie algebras this makes it possible to identify the given Lie algebra completely. The authors provide a representative list of all Lie algebras of dimension less or equal to 6 together with their important properties, including their Casimir invariants. The list is ordered in a way to make identification easy, using only basis independent properties of the Lie algebras. They also describe certain classes of nilpotent and solvable Lie algebras of arbitrary finite dimensions for which complete or partial classification exists and discuss in detail their construction and properties.
The book is based on material that was previously dispersed in journal articles, many of them written by one or both of the authors together with their collaborators. The reader of this book should be familiar with Lie algebra theory at an introductory level.
Contents
Title page
Contents
Preface
Acknowledgments
Part 1. General theory
Introduction and motivation
Basic concepts
Invariants of the coadjoint representation of a Lie algebra
Part 2. Recognition of a Lie algebra given by its structure constants
Identification of Lie algebras through the use of invariants
Decomposition into a direct sum
Levi decomposition. Identification of the radical and Levi factor
The nilradical of a Lie algebra
Part 3. Nilpotent, solvable and Levi decomposable Lie algebras
Nilpotent Lie algebras
Solvable Lie algebras and their nilradicals
Solvable Lie algebras with abelian nilradicals
Solvable Lie algebras with Heisenberg nilradical
Solvable Lie algebras with Borel nilradicals
Solvable Lie algebras with filiform and quasifiliform nilradicals
Levi decomposable algebras
Part 4. Low-dimensional Lie algebras
Structure of the lists of low-dimensional Lie algebras
Lie algebras up to dimension 3
Four-dimensional Lie algebras
Five-dimensional Lie algebras
Six-dimensional Lie algebras
Bibliography
Index
