Curtis Matrix Groups

1979
ISBN: 978-1-4684-0093-9
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 191 Seiten, eBook

Reihe: Universitext

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 isomorphie. 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 E of quaternions and note that for A E Mn(E) to operate linearlyon Rn we must operate on the right (since we multiply a vector by a scalar n n on the left). So we use row vectors for Rn, c E and write xA , for the row vector obtained by matrix multiplication. We get a complex-valued determinant function on Mn (E) such that det A # 0 guarantees that A has an inverse.

Zielgruppe

Graduate

Autoren/Hrsg.

Curtis, M. L.

Inhaltsverzeichnis

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 Rn.- 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 Rn (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.