Matrix Groups / Edition 2

Matrix Groups / Edition 2

by M. L. Curtis
Pub. Date:
Springer New York


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Matrix Groups / Edition 2

These notes were developed from a course taught at Rice University in the spring of 1976 and again at the University of Hawaii in the spring of 1977. It is assumed that the students know some linear algebra and a little about differentiation of vector-valued functions. The idea is to introduce some students to some of the concepts of Lie group theory —all done at the concrete level of matrix groups.

Product Details

ISBN-13: 9780387960746
Publisher: Springer New York
Publication date: 10/31/1984
Series: Universitext
Edition description: 2nd ed. 1984
Pages: 228
Product dimensions: 6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

1 General Linear Groups.- A. Groups.- B. Fields, Quaternions.- C. Vectors and Matrices.- D. General Linear Groups.- E. Exercises.- 2 Orthogonal Groups.- A. Inner Products.- B. Orthogonal Groups.- C. The Isomorphism Question.- D. Reflections in—n.- E. Exercises.- 3 Homomorphisms.- A. Curves in a Vector Space.- B. Smooth Homomorphisms.- C. Exercises.- 4 Exponential and Logarithm.- A. Exponential of a Matrix.- B. Logarithm.- C. One-parameter Subgroups.- D. Lie Algebras.- E. Exercises.- 5 SO(3) and Sp(1).- A. The Homomorphism—: S3?SO(3).- B. Centers.- C. Quotient Groups.- D. Exercises.- 6 Topology.- A. Introduction.- B. Continuity of Functions, Open Sets, Closed Sets.- C. Connected Sets, Compact Sets.- D. Subspace Topology, Countable Bases.- E. Manifolds.- F. Exercises.- 7 Maximal Tori.- A. Cartesian Products of Groups.- B. Maximal Tori in Groups.- C. Centers Again.- D. Exercises.- 8 Covering by Maximal Tori.- A. General Remarks.- B. (+) for U(n) and SU(n).- C. (+) for SO(n).- D. (+) for Sp(n).- E. Reflections in—n (again).- F. Exercises.- 9 Conjugacy of Maximal Tori.- A. Monogenic Groups.- B. Conjugacy of Maximal Tori.- C. The Isomorphism Question Again.- D. Simple Groups, Simply-Connected Groups.- E. Exercises.- 10 Spin(k).- A. Clifford Algebras.- B. Pin(k) and Spin(k).- C. The Isomorphisms.- D. Exercises.- 11 Normalizers, Weyl Groups.- A. Normalizers.- B. Weyl Groups.- C. Spin(2n+1) and Sp(n).- D. SO(n) Splits.- E. Exercises.- 12 Lie Groups.- A. Differentiable Manifolds.- B. Tangent Vectors, Vector Fields.- C. Lie Groups.- D. Connected Groups.- E. Abelian Groups.- 13.- A. Maximal Tori.- B. The Anatomy of a Reflection.- C. The Adjoint Representation.- D. Sample Computation of Roots.- Appendix 1.- Appendix 2.- References.- Supplementary Index (for Chapter 13).

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