Softcover ISBN:  9781470436544 
Product Code:  CRMM/33.S 
List Price:  $124.00 
MAA Member Price:  $111.60 
AMS Member Price:  $99.20 
Electronic ISBN:  9781470414726 
Product Code:  CRMM/33.E 
List Price:  $124.00 
MAA Member Price:  $111.60 
AMS Member Price:  $99.20 

Book DetailsCRM Monograph SeriesVolume: 33; 2014; 306 ppMSC: Primary 17; 81; 70; 37;
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 lowdimensional 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.ReadershipUndergraduate students, graduate students, and research mathematicians interested in structure and applications of Lie algebras.

Table of Contents

Chapters

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. Lowdimensional Lie algebras

Structure of the lists of lowdimensional Lie algebras

Lie algebras up to dimension 3

Fourdimensional Lie algebras

Fivedimensional Lie algebras

Sixdimensional Lie algebras


Additional Material

Reviews

Summarizing, this book is a highly welcome addition to the bookshelf and will certainly become a valuable and indispensable tool for the practitioner in Lie theory, as it presents in condensed form a huge quantity of information dispersed in the technical literature.
CMS Notes


RequestsReview Copy – for reviewers who would like to review an AMS bookAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
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 lowdimensional 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.
Undergraduate students, graduate students, and research mathematicians interested in structure and applications of Lie algebras.

Chapters

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. Lowdimensional Lie algebras

Structure of the lists of lowdimensional Lie algebras

Lie algebras up to dimension 3

Fourdimensional Lie algebras

Fivedimensional Lie algebras

Sixdimensional Lie algebras

Summarizing, this book is a highly welcome addition to the bookshelf and will certainly become a valuable and indispensable tool for the practitioner in Lie theory, as it presents in condensed form a huge quantity of information dispersed in the technical literature.
CMS Notes