Matrix Groups / Edition 2

by M. L. Curtis

Overview

These notes were developed from a course taught at Rice Univ- sity 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 students to some of the concepts of Lie group theory— all done at the concrete level of matrix groups. As much as we could, we motivated developments as a means of deciding when two matrix groups (with different definitions) are isomorphic. In Chapter I "group" is defined and examples are given; ho- morphism and isomorphism are defined. For a field k denotes the algebra of n x n matrices over k We recall that A E Mn(k) has an inverse if and only if det A ~ 0 , and define the general linear group GL(n,k) We construct the skew-field lli of to operate linearly on llin quaternions and note that for A E Mn(lli) we must operate on the right (since we mUltiply a vector by a scalar n on the left). So we use row vectors for R , en, llin and write xA for the row vector obtained by matrix multiplication. We get a ~omplex-valued determinant function on Mn (11) such that det A ~ 0 guarantees that A has an inverse.

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).