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.

lgrsnf/[CRM Monograph Series 33]Classification and Identification of Lie Algebras, Libor Šnobl, Pavel Winternitz, 2014, 306p, American Mathematical Society, 9780821843550.pdf

zlib/Mathematics/Libor Šnobl, Pavel Winternitz/Classification and Identification of Lie Algebras_6057398.pdf

Snobl, Libor; Winternitz, Pavel

Libor Snobl, Pavel Winternitz

American Mathematical Society, Providence, Rhode Island, 2014

Centre de Recherches Mathématiques, Providence, RI, 2014

United States, United States of America

lg2813792

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Includes bibliographical references (pages 299-304) and index.

Front Matter
Copyright
Contents
Preface
Acknowledgements
Part 1. General Theory
Chapter 1. Introduction and Motivation
Chapter 2. Basic Concepts
2.1. Definitions
2.2. Levi theorem
2.3. Classification of complex simple Lie algebras
2.4. Chevalley cohomology of Lie algebras
Chapter 3. Invariants of the Coadjoint Representation of a Lie Algebra
3.1. Casimir operators and generalized Casimir invariants
3.2. Calculation of generalized Casimir invariants using the infinitesimal method
3.2.1. Formulation of the problem and number of functionally independent solutions.
3.2.2. Method of characteristics for a single PDE.
3.2.3. Solution of the system of PDEs (3.2).
3.3. Calculation of generalized Casimir invariants by the method of moving frames
Part 2. Recognition of a Lie Algebra Given by Its Structure Constants
Chapter 4. Identification of Lie Algebras through the Use of Invariants
4.1. Elementary invariants
4.2. More sophisticated invariants
Chapter 5. Decomposition into a Direct Sum
5.1. General theory and criteria
5.2. Algorithm
5.3. Examples
Chapter 6. Levi Decomposition. Identification of the Radicaland Levi Factor
6.1. Original algorithm
6.2. Modified algorithm
6.3. Examples
Chapter 7. The Nilradical of a Lie Algebra
7.1. General theory
7.2. Algorithm
7.3. Examples
7.4. Identification of the nilradical using the Killing form
Part 3. Nilpotent, Solvable and Levi Decomposable Lie Algebras
Chapter 8. Nilpotent Lie Algebras
8.1. Maximal Abelian ideals and their extensions
8.2. Classification of low-dimensional nilpotent Lie algebras
Chapter 9. Solvable Lie Algebras and Their Nilradicals
9.1. General structure of a solvable Lie algebra
9.2. General procedure for classifying all solvable Lie algebras with a given nilradical
9.3. Upper bound on the dimension of a solvable extension of a given nilradical
9.4. Particular classes of nilradicals and their solvable extensions
9.5. Vector fields realizing bases of the coadjoint representation of a solvable Lie algebra
Chapter 10. Solvable Lie Algebras with Abelian Nilradicals
10.1. Basic structural theorems
10.2. Decomposability properties of the solvable Lie algebras
10.2.1. Nilradicals of minimal dimension.
10.3. Solvable Lie algebras with centers of maximal dimension
10.4. Solvable Lie algebras with one nonnilpotent element and an n-dimensional Abelian nilradical
10.5. Solvable Lie algebras with two nonnilpotent elements and n-dimensional Abelian nilradical
10.6. Generalized Casimir invariants of solvable Lie algebras with Abelian nilradicals
10.6.1. General form of the generalized Casimir invariants and their number.
10.6.2. Diagonal structure matrices.
10.6.3. Casimir invariants in the case f = 1.
10.6.4. Casimir invariants for low-dimensional nilradicals.
10.6.5. Summary.
Chapter 11. Solvable Lie Algebras with Heisenberg Nilradical
11.1. The Heisenberg relations and the Heisenberg algebra
11.2. Classification of solvable Lie algebras with nilradical h(m)
11.2.1. Basic classification theorem.
11.3. The lowest dimensional case m = 1
11.4. The case m = 2
11.5. Generalized Casimir invariants
Chapter 12. Solvable Lie Algebras with Borel Nilradicals
12.1. Outer derivations of nilradicals of Borel subalgebras
12.2. Solvable extensions of the Borel nilradicals NR(b(g))
12.2.1. Solvable extensions of the Borel nilradicals of maximal dimension.
12.2.2 Solvable extensions of the split real form of the nilradical NR(b(g))
12.2.3. Solvable extensions of the Borel nilradicals of less than maximal dimension.
12.2.4. Solvable extensions of dimension n_{NR}+1.
12.3. Solvable Lie algebras with triangular nilradicals
12.3.1. The structure of the algebras of strictly upper triangular matrices and their derivations.
12.3.2. Illustration of the procedure for low dimensions.
12.4. Casimir invariants of nilpotent and solvable triangular Lie algebras
12.4.1. Invariants of nilpotent triangular Lie algebras.
12.4.2. Invariants of the solvable triangular Lie algebras.
12.4.3. Casimir invariants of solvable extensions of t(4).
12.4.4. General results.
12.4.5. Conclusions.
Chapter 13. Solvable Lie Algebras with Filiform and Quasifiliform Nilradicals
13.1. Classification of solvable Lie algebras with the model filiform nilradical n_{n,1}
13.1.1. Nilpotent algebra n_{n,1}.
13.1.2. Construction of solvable Lie algebras with the nilradical n_{n,1}.
13.1.3. Standard forms of solvable Lie algebras with the nilradical n_{n,1}.
13.1.4. The generalized Casimir invariants of n_{n,1} and of its solvable extensions.
13.2. Classification of solvable Lie algebras with the nilradical n_{n,2}
13.2.1. Nilpotent algebra n_{n,2} and its structure.
13.2.2. Construction of solvable Lie algebras with the nilradical n_{n,2}.
13.2.3. Generalized Casimir invariants of n_{n,2} and of its solvable extension.
13.3. Solvable Lie algebras with other filiform nilradicals
13.4. Example of an almost filiform nilradical
13.4.1. Automorphisms and derivations of the nilradical n_{n,3}.
13.4.2. Construction of solvable Lie algebras with the nilradical n_{n,3}.
13.4.3. Dimension n = 6.
13.4.4. Dimension n = 5.
13.5. Generalized Casimir invariants of nn,3 and of its solvable extensions
Chapter 14. Levi Decomposable Algebras
14.1. Levi decomposable algebras with a nilpotent radical
14.2. Levi decomposable algebras with nonnilpotent radicals
14.3. Levi decomposable algebras of low dimensions
14.3.1. Levi extensions of Abelian radicals.
14.3.2. Levi extensions of non-Abelian radicals with Abelian nilradicals.
14.3.3. Levi decomposable algebras with non-Abelian nilradicals.
Part 4. Low-Dimensional Lie Algebras
Chapter 15. Structure of the Lists ofLow-Dimensional Lie Algebras
15.1. Ordering of the lists
15.2. Computer-assisted identification of a given Lie algebra
Chapter 16. Lie Algebras up to Dimension 3
16.1. One-dimensional Lie algebra
16.2. Solvable two-dimensional Lie algebra with the nilradical n_{1,1}
16.3. Nilpotent three-dimensional Lie algebra
16.4. Solvable three-dimensional Lie algebras with the nilradical 2n_{1,1}
16.5. Simple three-dimensional Lie algebras
Chapter 17. Four-Dimensional Lie Algebras
17.1. Nilpotent four-dimensional Lie algebra
17.2. Solvable four-dimensional algebras with the nilradical 3n_{1,1}
17.3. Solvable four-dimensional Lie algebras with the nilradical n_{3,1}
17.4. Solvable four-dimensional Lie algebras with the nilradical 2n_{1,1}
Chapter 18. Five-Dimensional Lie Algebras
18.1. Nilpotent five-dimensional Lie algebras
18.2. Solvable five-dimensional Lie algebras with the nilradical 4n_{1,1}
18.3. Solvable five-dimensional Lie algebras with the nilradical n_{3,1} ⊕ n_{1,1}
18.4. Solvable five-dimensional Lie algebras with the nilradical n_{4,1}
18.5. Solvable five dimensional Lie algebras with the nilradical 3n_{1,1}
18.6. Solvable five-dimensional Lie algebras with the nilradical n_{3,1}
Chapter 19. Six-Dimensional Lie Algebras
19.1. Nilpotent six-dimensional Lie algebras
19.2. Solvable six-dimensional Lie algebras with the nilradical 5n_{1,1}
19.3. Solvable six-dimensional Lie algebras with the nilradical n_{3,1} ⊕ 2n_{1,1}
19.4. Solvable six-dimensional Lie algebras with the nilradical n_{4,1} ⊕ n_{1,1}
19.5. Solvable six-dimensional Lie algebras with the nilradical n_{5,1}
19.6. Solvable six-dimensional Lie algebras with the nilradical n_{5,2}
19.7. Solvable six-dimensional Lie algebras with the nilradical n_{5,3}
19.8. Solvable six-dimensional Lie algebras with the nilradical n_{5,4}
19.9. Solvable six-dimensional Lie algebras with the nilradical n_{5,5}
19.10. Solvable six-dimensional Lie algebra with the nilradical n_{5,6}
19.11. Solvable six-dimensional Lie algebras with the nilradical 4n_{1,1}
19.12. Solvable six-dimensional Lie algebras with the nilradical n_{3,1} ⊕ n_{1,1}
19.13. Solvable six-dimensional Lie algebra with the nilradical n_{4,1}
Bibliography
Index
Book Presentation by Authors
The monograph: Classification and Identification of Lie Algebras
Why to be interested in identification and classification of Lie algebras?
What can be found in our book
Conclusions

Cover -- 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 -- Back Cover

2020-10-21