Lie Groups

by Pollatsek, Harriet

• ISBN13: 9780883857595
• ISBN10: 0883857596
• Format: Hardcover
• Copyright: 2009-08-31
• Publisher: Mathematical Assn of Amer

Summary

This textbook is a complete introduction to Lie groups for undergraduate students. The only prerequisites are multi-variable calculus and linear algebra. The emphasis is placed on the algebraic ideas, with just enough analysis to define the tangent space and the differential and to make sense of the exponential map. This textbook works on the principle that students learn best when they are actively engaged. To this end nearly 200 problems are included in the text, ranging from the routine to the challenging level. Every chapter has a section called 'Putting the pieces together' in which all definitions and results are collected for reference and further reading is suggested.

Table of Contents

Symmetries of vector spaces

What is a symmetry?

Distance is fundamental

Groups of symmetries

Bilinear forms and symmetries of spacetime

Putting the pieces together

A broader view: Lie groups

Complex numbers, quaternions and geometry

Complex numbers

Quaternions

The geometry of rotations of R3

Putting the pieces together

A broader view: octonions

Linearization

Tangent spaces

Group homomorphisms

Differentials

Putting the pieces together

A broader view: Hilbert's fifth problem

One-parameter subgroups and the exponential map

One-parameter subgroups

The exponential map in dimension one

Calculating the matrix exponential

Properties of the matrix exponential

Using exp to determine L(G)

Differential equations

Putting the pieces together

A broader view: Lie and differential equations

Appendix on convergence

Lie algebras

Lie algebras

Adjoint maps { big `A' and small `a'

Putting the pieces together

A broader view: Lie theory

Matrix groups over other fields

What is a field?

The unitary group

Matrix groups over finite fields

Putting the pieces together

A broader view of finite groups of Lie type and simple groups

Linear algebra facts

Paper assignment used at Mount Holyoke College

Opportunities for further study

Solutions to selected problems

Bibliography

Table of Contents provided by Publisher. All Rights Reserved.