Matrix Groups

by Baker, Andrew

• ISBN13: 9781852334703
• ISBN10: 1852334703
• Format: Paperback
• Copyright: 2002-01-01
• Publisher: Springer Verlag

Summary

Aimed at advanced undergraduate and beginning graduate students, this book provides a first taste of the theory of Lie groups as an appetiser for a more substantial further course. Lie theoretic ideas lie at the heart of much of standard undergraduate linear algebra and exposure to them can inform or motivate the study of the latter.The main focus is on matrix groups, i.e., closed subgroups of real and complex general linear groups. The first part studies examples and describes the classical families of simply connected compact groups. The second part introduces the idea of a lie group and studies the associated notion of a homogeneous space using orbits of smooth actions.Throughout, the emphasis is on providing an approach that is accessible to readers equipped with a standard undergraduate toolkit of algebra and analysis. Although the formal prerequisites are kept as low level as possible, the subject matter is sophisticated and contains many of the key themes of the fully developed theory, preparing students for a more standard and abstract course in Lie theory and differential geometry.

Table of Contents

Part I. Basic Ideas and Examples
Real and Complex Matrix Groups	3	(42)
Groups of Matrices	3	(2)
Groups of Matrices as Metric Spaces	5	(7)
Compactness	12	(3)
Matrix Groups	15	(2)
Some Important Examples	17	(12)
Complex Matrices as Real Matrices	29	(2)
Continuous Homomorphisms of Matrix Groups	31	(2)
Matrix Groups for Normed Vector Spaces	33	(4)
Continuous Group Actions	37	(8)
Exponentials, Differential Equations and One-parameter Subgroups	45	(22)
The Matrix Exponential and Logarithm	45	(6)
Calculating Exponentials and Jordan Form	51	(4)
Differential Equations in Matrices	55	(1)
One-parameter Subgroups in Matrix Groups	56	(3)
One-parameter Subgroups and Differential Equations	59	(8)
Tangent Spaces and Lie Algebras	67	(32)
Lie Algebras	67	(4)
Curves, Tangent Spaces and Lie Algebras	71	(5)
The Lie Algebras of Some Matrix Groups	76	(8)
Some Observations on the Exponential Function of a Matrix Group	84	(2)
SO(3) and SU(2)	86	(6)
The Complexification of a Real Lie Algebra	92	(7)
Algebras, Quaternions and Quaternionic Sympletic Groups	99	(30)
Algebras	99	(12)
Real and Complex Normed Algebras	111	(2)
Linear Algebra over a Division Algebra	113	(3)
The Quaternions	116	(4)
Quaternionic Matrix Groups	120	(2)
Automorphism Groups of Algebras	122	(7)
Clifford Algebras and Spinor Groups	129	(28)
Real Clifford Algebras	130	(9)
Clifford Groups	139	(4)
Pinor and Spinor Groups	143	(8)
The Centres of Spinor Groups	151	(1)
Finite Subgroups of Spinor Groups	152	(5)
Lorentz Groups	157	(24)
Lorentz Groups	157	(8)
A Principal Axis Theorem for Lorentz Groups	165	(6)
SL2(C) and the Lorentz Group Lor(3, 1)	171	(10)
Part II. Matrix Groups as Lie Groups
Lie Groups	181	(30)
Smooth Manifolds	181	(2)
Tangent Spaces and Derivatives	183	(4)
Lie Groups	187	(2)
Some Examples of Lie Groups	189	(4)
Some Useful Formulae in Matrix Groups	193	(6)
Matrix Groups are Lie Groups	199	(4)
Not All Lie Groups are Matrix Groups	203	(8)
Homogeneous Spaces	211	(24)
Homogeneous Spaces as Manifolds	211	(4)
Homogeneous Spaces as Orbits	215	(2)
Projective Spaces	217	(5)
Grassmannians	222	(2)
The Gram-Schmidt Process	224	(2)
Reduced Echelon Form	226	(1)
Real Inner Products	227	(2)
Symplectic Forms	229	(6)
Connectivity of Matrix Groups	235	(16)
Connectivity of Manifolds	235	(3)
Examples of Path Connected Matrix Groups	238	(3)
The Path Components of a Lie Group	241	(3)
Another Connectivity Result	244	(7)
Part III. Compact Connected Lie Groups and their Classification
Maximal Tori in Compact Connected Lie Groups	251	(16)
Tori	251	(4)
Maximal Tori in Compact Lie Groups	255	(4)
The Normaliser and Weyl Group of a Maximal Torus	259	(3)
The Centre of a Compact Connected Lie Group	262	(5)
Semi-simple Factorisation	267	(22)
An Invariant Inner Product	267	(3)
The Centre and its Lie Algebra	270	(2)
Lie Ideals and the Adjoint Action	272	(4)
Semi-simple Decompositions	276	(2)
The Structure of the Adjoint Representation	278	(11)
Roots Systems, Weyl Groups and Dynkin Diagrams	289	(14)
Inner Products and Duality	289	(2)
Roots systems and their Weyl groups	291	(2)
Some Examples of Root Systems	293	(4)
The Dynkin Diagram of a Root System	297	(1)
Irreducible Dynkin Diagrams	298	(1)
From Root Systems to Lie Algebras	299	(4)
Hints and Solutions to Selected Exercises	303	(20)
Bibliography	323	(2)
Index	325