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Classification and Identification of Lie Algebras [Kietas viršelis]

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  • Formatas: Hardback, 306 pages, aukštis x plotis: 229x152 mm
  • Serija: CRM Monograph Series
  • Išleidimo metai: 01-Jan-2014
  • Leidėjas: American Mathematical Society
  • ISBN-10: 0821843559
  • ISBN-13: 9780821843550
Kitos knygos pagal šią temą:
Classification and Identification of Lie AlgebrasDidesnis paveikslėlis
  • Formatas: Hardback, 306 pages, aukštis x plotis: 229x152 mm
  • Serija: CRM Monograph Series
  • Išleidimo metai: 01-Jan-2014
  • Leidėjas: American Mathematical Society
  • ISBN-10: 0821843559
  • ISBN-13: 9780821843550
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780821843550.html
Raktažodžiai:
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.
Preface ix
Acknowledgements xi
Part
1. General Theory		 1(36)
Chapter 1 Introduction and Motivation
3(8)
Chapter 2 Basic Concepts
11(12)
2.1 Definitions
11(6)
2.2 Levi theorem
17(1)
2.3 Classification of complex simple Lie algebras
17(3)
2.4 Chevalley cohomology of Lie algebras
20(3)
Chapter 3 Invariants of the Coadjoint Representation of a Lie Algebra
23(14)
3.1 Casimir operators and generalized Casimir invariants
23(1)
3.2 Calculation of generalized Casimir invariants using the infinitesimal method
24(8)
3.3 Calculation of generalized Casimir invariants by the method of moving frames
32(5)
Part
2. Recognition of a Lie Algebra Given by Its Structure Constants		 37(50)
Chapter 4 Identification of Lie Algebras through the Use of Invariants
39(8)
4.1 Elementary invariants
39(3)
4.2 More sophisticated invariants
42(5)
Chapter 5 Decomposition into a Direct Sum
47(16)
5.1 General theory and criteria
47(9)
5.2 Algorithm
56(1)
5.3 Examples
57(6)
Chapter 6 Levi Decomposition. Identification of the Radical and Levi Factor
63(8)
6.1 Original algorithm
63(2)
6.2 Modified algorithm
65(1)
6.3 Examples
66(5)
Chapter 7 The Nilradical of a Lie Algebra
71(16)
7.1 General theory
71(4)
7.2 Algorithm
75(4)
7.3 Examples
79(5)
7.4 Identification of the nilradical using the Killing form
84(3)
Part
3. Nilpotent, Solvable and Levi Decomposable Lie Algebras		 87(128)
Chapter 8 Nilpotent Lie Algebras
89(10)
8.1 Maximal Abelian ideals and their extensions
89(4)
8.2 Classification of low-dimensional nilpotent Lie algebras
93(6)
Chapter 9 Solvable Lie Algebras and Their Nilradicals
99(8)
9.1 General structure of a solvable Lie algebra
99(1)
9.2 General procedure for classifying all solvable Lie algebras with a given nilradical
99(4)
9.3 Upper bound on the dimension of a solvable extension of a given nilradical
103(2)
9.4 Particular classes of nilradicals and their solvable extensions
105(1)
9.5 Vector fields realizing bases of the coadjoint representation of a solvable Lie algebra
106(1)
Chapter 10 Solvable Lie Algebras with Abelian Nilradicals
107(24)
10.1 Basic structural theorems
107(7)
10.2 Decomposability properties of the solvable Lie algebras
114(2)
10.3 Solvable Lie algebras with centers of maximal dimension
116(5)
10.4 Solvable Lie algebras with one nonnilpotent element and an n-dimensional Abelian nilradical
121(2)
10.5 Solvable Lie algebras with two nonnilpotent elements and n-dimensional Abelian nilradical
123(2)
10.6 Generalized Casimir invariants of solvable Lie algebras with Abelian nilradicals
125(6)
Chapter 11 Solvable Lie Algebras with Heisenberg Nilradical
131(10)
11.1 The Heisenberg relations and the Heisenberg algebra
131(1)
11.2 Classification of solvable Lie algebras with nilradical h(m)
132(2)
11.3 The lowest dimensional case m = 1
134(1)
11.4 The case m = 2
135(1)
11.5 Generalized Casimir invariants
136(5)
Chapter 12 Solvable Lie Algebras with Borel Nilradicals
141(34)
12.1 Outer derivations of nilradicals of Borel subalgebras
141(5)
12.2 Solvable extensions of the Borel nilradicals NR(b(g))
146(7)
12.3 Solvable Lie algebras with triangular nilradicals
153(9)
12.4 Casimir invariants of nilpotent and solvable triangular Lie algebras
162(13)
Chapter 13 Solvable Lie Algebras with Filiform and Quasifiliform Nilradicals
175(28)
13.1 Classification of solvable Lie algebras with the model filiform nilradical nn,1
176(6)
13.2 Classification of solvable Lie algebras with the nilradical nn,2
182(7)
13.3 Solvable Lie algebras with other filiform nilradicals
189(1)
13.4 Example of an almost filiform nilradical
190(9)
13.5 Generalized Casimir invariants of nn,3 and of its solvable extensions
199(4)
Chapter 14 Levi Decomposable Algebras
203(12)
14.1 Levi decomposable algebras with a nilpotent radical
204(3)
14.2 Levi decomposable algebras with nonnilpotent radicals
207(1)
14.3 Levi decomposable algebras of low dimensions
208(7)
Part
4. Low-Dimensional Lie Algebras		 215(84)
Chapter 15 Structure of the Lists of Low-Dimensional Lie Algebras
217(8)
15.1 Ordering of the lists
217(1)
15.2 Computer-assisted identification of a given Lie algebra
218(7)
Chapter 16 Lie Algebras up to Dimension 3
225(2)
16.1 One-dimensional Lie algebra
225(1)
16.2 Solvable two-dimensional Lie algebra with the nilradical n1,1
225(1)
16.3 Nilpotent three-dimensional Lie algebra
225(1)
16.4 Solvable three-dimensional Lie algebras with the nilradical 2n1,1
226(1)
16.5 Simple three-dimensional Lie algebras
226(1)
Chapter 17 Four-Dimensional Lie Algebras
227(4)
17.1 Nilpotent four-dimensional Lie algebra
227(1)
17.2 Solvable four-dimensional algebras with the nilradical 3n1,1
227(1)
17.3 Solvable four-dimensional Lie algebras with the nilradical n3,1
228(1)
17.4 Solvable four-dimensional Lie algebras with the nilradical 2n1,1
229(2)
Chapter 18 Five-Dimensional Lie Algebras
231(12)
18.1 Nilpotent five-dimensional Lie algebras
231(1)
18.2 Solvable five-dimensional Lie algebras with the nilradical 4n1,1
232(3)
18.3 Solvable five-dimensional Lie algebras with the nilradical n3,1 n1,1
235(4)
18.4 Solvable five-dimensional Lie algebras with the nilradical n4,1
239(1)
18.5 Solvable five dimensional Lie algebras with the nilradical 3n1,1
240(1)
18.6 Solvable five-dimensional Lie algebras with the nilradical n3,1
241(1)
18.7 Five-dimensional Levi decomposable Lie algebra
241(2)
Chapter 19 Six-Dimensional Lie Algebras
243(56)
19.1 Nilpotent six-dimensional Lie algebras
243(5)
19.2 Solvable six-dimensional Lie algebras with the nilradical 5n1,1
248(5)
19.3 Solvable six-dimensional Lie algebras with the nilradical n3,1 2n1,1
253(13)
19.4 Solvable six-dimensional Lie algebras with the nilradical n4,1 n1,1
266(5)
19.5 Solvable six-dimensional Lie algebras with the nilradical n5,1
271(6)
19.6 Solvable six-dimensional Lie algebras with the nilradical n5,2
277(2)
19.7 Solvable six-dimensional Lie algebras with the nilradical n5,3
279(4)
19.8 Solvable six-dimensional Lie algebras with the nilradical n5,4
283(2)
19.9 Solvable six-dimensional Lie algebras with the nilradical n5,5
285(1)
19.10 Solvable six-dimensional Lie algebra with the nilradical n5,6
286(1)
19.11 Solvable six-dimensional Lie algebras with the nilradical 4n1,1
286(7)
19.12 Solvable six-dimensional Lie algebras with the nilradical n3,1 n1,1
293(3)
19.13 Solvable six-dimensional Lie algebra with the nilradical n4,1
296(1)
19.14 Simple six-dimensional Lie algebra
296(1)
19.15 Six-dimensional Levi decomposable Lie algebras
296(3)
Bibliography 299(6)
Index 305
Libor nobl, Czech Technical University, Prague, Czech Republic

Pavel Winternitz, Centre de Recherches Mathématiques, Montréal, QC, Canada, and Université de Montréal, QC, Canada