**CRM Monograph Series**

Volume: 33;
2014;
306 pp;
Hardcover

MSC: Primary 17; 81; 70; 37;

Print ISBN: 978-0-8218-4355-0

Product Code: CRMM/33

List Price: $124.00

Individual Member Price: $99.20

**Electronic ISBN: 978-1-4704-1472-6
Product Code: CRMM/33.E**

List Price: $124.00

Individual Member Price: $99.20

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#### Supplemental Materials

# Classification and Identification of Lie Algebras

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*Libor Šnobl; Pavel Winternitz*

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.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Table of Contents

# Table of Contents

## Classification and Identification of Lie Algebras

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Acknowledgments xi12
- Part 1. General theory 114
- Introduction and motivation 316
- Basic concepts 1124
- Invariants of the coadjoint representation of a Lie algebra 2336
- Part 2. Recognition of a Lie algebra given by its structure constants 3750
- Identification of Lie algebras through the use of invariants 3952
- Decomposition into a direct sum 4760
- Levi decomposition. Identification of the radical and Levi factor 6376
- The nilradical of a Lie algebra 7184
- Part 3. Nilpotent, solvable and Levi decomposable Lie algebras 87100
- Nilpotent Lie algebras 89102
- Solvable Lie algebras and their nilradicals 99112
- Solvable Lie algebras with abelian nilradicals 107120
- Solvable Lie algebras with Heisenberg nilradical 131144
- Solvable Lie algebras with Borel nilradicals 141154
- Solvable Lie algebras with filiform and quasifiliform nilradicals 175188
- Levi decomposable algebras 203216
- Part 4. Low-dimensional Lie algebras 215228
- Structure of the lists of low-dimensional Lie algebras 217230
- Lie algebras up to dimension 3 225238
- Four-dimensional Lie algebras 227240
- Five-dimensional Lie algebras 231244
- Six-dimensional Lie algebras 243256
- Bibliography 299312
- Index 305318
- Back Cover Back Cover1321

#### Readership

Undergraduate students, graduate students, and research mathematicians interested in structure and applications of 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