Lie Groups, Lie Algebras, and Some of Their Applications

by Gilmore, Robert

• ISBN13: 9780486445298
• ISBN10: 0486445291
• Format: Paperback
• Copyright: 2006-01-04
• Publisher: Dover Publications

Summary

With rigor and clarity, this upper-level undergraduate text employs numerous exercises, solved problems, and figures to introduce upper-level undergraduates to Lie group theory and physical applications. It further illustrates Lie group theory's role in expressing concepts and results from several fields of physics. 1974 edtion. Includes 75 figures and 17 tables.

Table of Contents

1 Introductory Concepts	1	(23)
I. BASIC BUILDING BLOCKS,	1	(9)
1. Set,	1	(1)
2. Group,	1	(2)
3. Field,	3	(2)
4. Linear Vector Space,	5	(3)
5. Algebra,	8	(2)
II. BASES,	10	(3)
1. For a Group,	11	(1)
2. For a Field,	12	(1)
3. For a Vector Space,	12	(1)
4. For an Algebra,	12	(1)
III. MAPPINGS, REALIZATIONS, REPRESENTATIONS,	13	(6)
1. Sets,	14	(1)
2. Groups,	15	(1)
3. Fields,	16	(2)
4. Vector Spaces,	18	(1)
5. Algebras,	19	(1)
RÉSUMÉ,	19	(1)
EXERCISES,	20	(3)
NOTES AND REFERENCES,	23	(1)
2 The Classical Groups	24	(33)
I. GENERAL LINEAR GROUPS,	24	(2)
1. Change of Basis,	24	(1)
2. Covariance and Contravariance,	25	(1)
II. VOLUME PRESERVING GROUPS,	26	(11)
1. Direct Sum,	26	(2)
2. Direct Product,	28	(2)
3. Symmetric Reduction in Tensor Space,	30	(3)
4. Fully Symmetric Subspaces—Expansion of Functions,	33	(3)
5. Fully Antisymmetric Subspaces—Volume Element,	36	(1)
III. METRIC PRESERVING GROUPS,	37	(10)
1. The Metric,	37	(2)
2. Kinds of Metrics,	39	(3)
3. Weyl Unitary Trick,	42	(1)
4. Metrics in Function Spaces,	43	(1)
5. Metric Preserving Groups,	43	(4)
IV. PROPERTIES OF THE CLASSICAL GROUPS,	47	(5)
1. Relationships Among the Classical Groups,	47	(2)
2. Dimensions of the Classical Groups,	49	(3)
3. Isomorphisms and Homomorphisms Among the Classical Groups,	52	(1)
RÉSUMÉ,	52	(1)
EXERCISES,	53	(3)
NOTES AND REFERENCES,	56	(1)
3 Continuous Groups—Lie Groups	57	(30)
I. TOPOLOGICAL GROUPS,	57	(9)
1. Some Basic Definitions,	57	(8)
2. Comments,	65	(1)
3. Additional Comments,	66	(1)
II. AN EXAMPLE,	66	(5)
1. The Two-Parameter Group of Transformations on the Straight Line R1,	66	(3)
2. Some Realizations for This Group,	69	(2)
III. ADDITIONAL NECESSARY CONCEPTS,	71	(6)
1. Topological Concepts,	71	(1)
2. Algebraic Concepts,	72	(4)
3. Local Concepts,	76	(1)
IV. LIE GROUPS,	77	(1)
1. The Motivation,	77	(1)
V. THE INVARIANT INTEGRAL,	78	(6)
1. The Rearrangement Property,	78	(1)
2. Reparameterization of G0,	79	(1)
3. General Left and Right Invariant Densities,	80	(1)
4. Equality of Left and Right Measures,	81	(3)
5. Extension to Continuous Groups,	84	(1)
RÉSUMÉ,	84	(1)
EXERCISES,	84	(2)
NOTES AND REFERENCES,	86	(1)
4 Lie Groups and Lie Algebras	87	(33)
I. INFINITESIMAL PROPERTIES OF LIE GROUPS,	87	(9)
1. Infinitesimal Generators for Lie Groups of Transformations,	87	(4)
2. Infinitesimal Generators for a Lie Group,	91	(1)
3. Infinitesimal Generators for Matrix Groups,	92	(2)
4. Commutation Relations,	94	(2)
II. LIE'S FIRST THEOREM,	96	(4)
1. Theorem,	96	(2)
2. Example,	98	(1)
3. Comment,	99	(1)
III. LIE'S SECOND THEOREM,	100	(4)
1. Theorem,	100	(3)
2. Example,	103	(1)
3. Comment,	103	(1)
IV. LIE'S THIRD THEOREM,	104	(2)
1. Theorem,	104	(1)
2. Structure Constants as Matrix Elements,	105	(1)
V. CONVERSES OF LIE'S THREE THEOREMS,	106	(5)
1. The Converses,	107	(1)
2. Comments,	108	(1)
3. Example,	109	(1)
4. An Important Comment,	109	(2)
VI. TAYLOR'S THEOREM FOR LIE GROUPS,	111	(6)
1. The Theorem,	111	(2)
2. An Auxiliary Result,	113	(1)
3. Comment,	114	(1)
4. τ-Ordered Products,	114	(3)
RÉSUMÉ,	117	(1)
EXERCISES,	118	(1)
NOTES AND REFERENCES,	119	(1)
5 Some Simple Examples	120	(62)
I. RELATIONS AMONG SOME LIE ALGEBRAS,	121	(5)
1. 1 x 1 Quaternion Groups,	121	(1)
2. 2 x 2 Unitary Groups,	122	(2)
3. 3 x 3 Orthogonal Groups,	124	(2)
II. COMPARISON OF LIE GROUPS,	126	(9)
1. Parameter Spaces for S1(1, q), SU(2, c), SO(3, r),	126	(3)
2. Connectivity,	129	(4)
3. Homotopy and Discrete Invariant Subgroups,	133	(2)
III. REPRESENTATIONS OF SU(2, c),	135	(7)
1. General Considerations,	135	(1)
2. Tensor Product Representations,	136	(5)
3. Representations of SO(3, r),	141	(1)
IV. QUATERNION COVERING GROUP,	142	(4)
1. Direct Sums and Products,	142	(1)
2. Representations of the Factor Groups,	142	(3)
3. Two Possible Physical Consequences,	145	(1)
V. SPIN AND DOUBLE-VALUEDNESS-DESCRIPTION OF THE ELECTRON,	146	(3)
VI. NONCANONICAL PARAMETERIZATIONS FOR SU(2; c),	149	(29)
1. Baker-Campbell-Hausdorff Formulas,	149	(4)
2. Application in Constructing Representations,	153	(5)
3. Physical Applications,	158	(20)
RÉSUMÉ,	178	(1)
EXERCISES,	179	(2)
NOTES AND REFERENCES,	181	(1)
6 Classical Algebras	182	(34)
I. COMPUTATION OF THE ALGEBRAS,	182	(17)
1. General Procedures,	182	(1)
2. Unitary Groups,	183	(3)
3. OfthogOnal Groups,	186	(1)
4. Symplectic Groups,	187	(2)
5. Bases for These Algebras,	189	(3)
6. Origin of the Embedding Groups SO*(2n) and SU*(20,	192	(5)
7. Summary of the Real Forms of the Classical Groups,	197	(2)
II. TOPOLOGICAL PROPERTIES,	199	(13)
1. Connectivity,	199	(1)
2. Cosets,	200	(9)
3. Contraction,	209	(3)
RÉSUMÉ,	212	(1)
EXERCISES,	213	(2)
NOTES AND REFERENCES,	215	(1)
7 Lie Algebras and Root Spaces	216	(62)
I. GENERAL STRUCTURE THEORY FOR LIE ALGEBRAS,	216	(21)
1. The Basic Tools,	216	(2)
2. The Regular Representation,	218	(2)
3. Systematics of Subalgebras,	220	(9)
4. Lie's Theorem,	229	(5)
5. Classification of Lie Algebras,	234	(3)
II. THE SECULAR EQUATION,	237	(11)
1. Rank,	237	(4)
2. Jordan Canonical Form,	241	(2)
3. First Criterion of Solvability,	243	(1)
4. Properties of the Root Subspaces,	244	(4)
III. THE METRIC,	248	(7)
1. Motivation for Choice,	248	(1)
2. Properties and Examples,	249	(2)
3. Metrics in Other Representations,	251	(2)
4. Second Criterion of Solvability,	253	(2)
IV. CARTAN'S CRITERION,	255	(6)
1. The Criterion,	255	(1)
2. Comments and Examples,	256	(3)
3. Complete Reducibility of Semisimple Algebras,	259	(2)
V. CANONICAL COMMUTATION RELATIONS FOR SEMISIMPLE ALGEBRAS,	261	(12)
1. Structure of the Metric in a Root Subspace Decomposition,	261	(2)
2. Properties of the Cartan Subalgebra,	263	(1)
3. First Chain Condition,	264	(1)
4. Second Chain Condition,	265	(5)
5. The Canonical Commutation Relations,	270	(3)
RÉSUMÉ,	273	(1)
EXERCISES,	274	(3)
NOTES AND REFERENCES,	277	(1)
8 Root Spaces and Dynkin Diagrams	278	(40)
I. CLASSIFICATION OF THE SIMPLE ROOT SPACES,	278	(15)
1. Review of the Canonical Commutation Relations,	278	(2)
2. The Rank-2 Root Spaces,	280	(5)
3. Construction of the Simple Root Spaces,	285	(3)
4. The E Series,	288	(2)
5. List of the Simple Root Spaces,	290	(3)
II. IDENTIFICATION OF THE CLASSICAL ALGEBRAS,	293	(5)
1. An and Sl(n + 1; c),	293	(1)
2. The Sibling Algebras Dn, Bn, Cn,	294	(4)
III. DYNKIN DIAGRAMS,	298	(17)
1. Simple Roots,	298	(8)
2. Properties of the Dynkin Diagrams,	306	(9)
RÉSUMÉ,	315	(1)
EXERCISES,	316	(1)
NOTES AND REFERENCES,	317	(1)
9 Real Forms	318	(118)
I. ALGEBRAIC MACHINERY,	319	(20)
1. Automorphisms,	319	(7)
2. Some Relations Between Algebraic and Topological Properties,	326	(8)
3. The Classification Machinery,	334	(5)
II. CLASSIFICATION OF THE REAL FORMS,	339	(7)
1. An-1,	339	(3)
2. Bn,	342	(1)
3. Dn,	342	(1)
4. Cn,	343	(2)
5. Exceptional Groups,	345	(1)
III. DISCUSSION OF RESULTS,	346	(4)
1. Tables of the Real Forms,	346	(2)
2. The Character Function,	348	(2)
IV. PROPERTIES OF COSETS,	350	(13)
1. Matrix Properties,	350	(3)
2. Inner Products and Index,	353	(1)
3. Rank,	354	(9)
V. ANALYTICAL PROPERTIES OF COSETS,	363	(43)
1. Structure of the Classical Cosets,	363	(3)
2. Global Transformation Properties,	366	(14)
3. Infinitesimal Transformations,	380	(3)
4. Infinitesimal's Transformations,	383	(2)
5. Geodesics, Distance, Metric,	385	(6)
6. Measure and Volume on Cosets,	391	(4)
7. Measure and Volume on the Classical Groups,	395	(6)
8. Canonical Coset Parameterization of the Classical Groups,	401	(5)
VI. REAL FORMS OF THE SYMMETRIC SPACES,	406	(16)
1. Algebraic Machinery,	406	(4)
2. Tables of the Real Forms,	410	(11)
3. Characters of the Real Forms,	421	(1)
RÉSUMÉ,	422	(1)
EXERCISES,	422	(13)
NOTES AND REFERENCES,	435	(1)
10 Contractions and Expansions	436	(71)
I. SIMPLE CONTRACTIONS,	436	(27)
1. The Little Prince,	436	(10)
2. Inönü-Wigner Contractions,	446	(4)
3. Some Useful Contractions,	450	(10)
4. The Baker-Campbell-Hausdorff Formula,	460	(3)
II. SALETAN CONTRACTIONS,	463	(14)
1. The First Contraction,	463	(6)
2. Further Contractions,	469	(3)
3. Saletan Contractions in a Kupczynski Basis,	472	(5)
III. EXPANSIONS,	477	(15)
1. Examples of Useful Expansions,	478	(5)
2. Expansions of Rank-1 Spaces,	483	(9)
RÉSUMÉ,	492	(1)
EXERCISES,	492	(13)
NOTES AND REFERENCES,	505	(2)
Bibliography	507	(58)
Author Index	565	(2)
Subject Index	567