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Classification and Identification of Lie Algebras

Libor Šnobl Czech Technical University, Prague, Czech Republic
Pavel Winternitz Centre de Recherches Mathématiques, Montréal, QC, Canada and Université de Montréal, Montréal, QC, Canada
A co-publication of the AMS and Centre de Recherches Mathématiques
Available Formats:
Softcover ISBN: 978-1-4704-3654-4
Product Code: CRMM/33.S
List Price: $124.00 MAA Member Price:$111.60
AMS Member Price: $99.20 Electronic ISBN: 978-1-4704-1472-6 Product Code: CRMM/33.E List Price:$124.00
MAA Member Price: $111.60 AMS Member Price:$99.20
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List Price: $186.00 MAA Member Price:$167.40
AMS Member Price: $148.80 Click above image for expanded view Classification and Identification of Lie Algebras Libor Šnobl Czech Technical University, Prague, Czech Republic Pavel Winternitz Centre de Recherches Mathématiques, Montréal, QC, Canada and Université de Montréal, Montréal, QC, Canada A co-publication of the AMS and Centre de Recherches Mathématiques Available Formats:  Softcover ISBN: 978-1-4704-3654-4 Product Code: CRMM/33.S  List Price:$124.00 MAA Member Price: $111.60 AMS Member Price:$99.20
 Electronic ISBN: 978-1-4704-1472-6 Product Code: CRMM/33.E
 List Price: $124.00 MAA Member Price:$111.60 AMS Member Price: $99.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$186.00 MAA Member Price: $167.40 AMS Member Price:$148.80
• Book Details

CRM Monograph Series
Volume: 332014; 306 pp
MSC: 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 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.

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. 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

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 332014; 306 pp
MSC: 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 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.

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. 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
• 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.