Naive Lie Theory

2009年Springer New York出版社出版John Stillwell編著圖書。

基本介紹

• 中文名稱：Naive Lie Theory
• 裝幀：Paperback
• 定價：USD 49.95
• 作者：John Stillwell
• 出版社：Springer New York
• 出版日期：2009-12-28
• ISBN：9781441926814

編輯推薦

《樸素李理論》是一部介紹李群和李代數的本科生教程，基本的微積分和線性代數知識將對理解《樸素李理論》十分重要。為了讓更多讀者受益，也是《樸素李理論》的最直接目的，書中對經典群核實、復和四元數空間做了較深刻地介紹。書中從矩陣的角度講述對稱群，這樣就可以用微積分和線性代數的基礎理論理解《樸素李理論》中的內容。《樸素李理論》由（美）史迪威著。

目錄

1 Geometry of complex numbers and quaternions

1.1 Rotations of the plane

1.2 Matrix representation of complex numbers

1.3 Quaternions

1.4 Consequences of multiplicative absolute value

1.5 Quaternion representation of space rotations

1.6 Discussion

2 Groups

2.1 Crash course on groups

2.2 Crash course on homomorphisms

2.3 The groups SU(2) and SO(3)

2.4 Isometrics of R'' and reflections

2.5 Rotations of R4 and pairs of quaternions

2.6 Direct products of groups

2.7 The map from SU(2)SU(2) to SO(4)

2.8 Discussion

3 Generalized rotation groups

3.1 Rotations as orthogonal transformations

3.2 The orthogonai and special orthogonal groups

3.3 The unitary groups

3.4 The symplectic groups

3.5 Maximal tori and centers

3.6 Maximal tori in SO(n), U(n), SU(n), Sp(n)

3.7 Centers of SO(n), U(n), SU(n), Sp(n)

3.8 Connectcdness and discreteness

3.9 Discussion

4 The exponential map

4.1 The exponential map onto SO(2)

4.2 The exponential map onto SU(2)

4.3 The tangent space of SU(2)

4.4 The Lie algebra su(2) of SU(2)

4.5 The exponential of a square matrix

4.6 The affine group of the line

4.7 Discussion

5 The tangent space

5.1 Tangent vectors of O(n), U(n), Sp(n)

5.2 The tangent space of SO(n)

5.3 The tangent space of U(n), SU(n), Sp(n)

5.4 Algebraic properties of the tangent space

5.5 Dimension of Lie algebras

5.6 Complexification

5.7 Quaternion Lie algebras

5.8 Discussion

6 Structure of Lie algebras

6.1 Normal subgroups and ideals

6.2 Ideals and homomorphisms

6.3 Classical non-simple Lie algebras

6.4 Simplicity of (n,C) and su(n)

6.5 Simplicity of o(n) for n > 4

6.6 Simplicity of p(n)

6.7 Discussion

7 The matrix logarithm

7.1 Logarithm and exponential

7.2 The exp function on the tangent space

7.3 Limit properties of log and exp

7.4 The log function into the tangentspace

7.5 SO(n), SU(n), and Sp(n) revisited

7.6 The Campbell-Baker-Hausdorff theorem

7.7 Eichler's proof of Campbell-Baker-Hausdorff

7,8 Discussion

8 Topology

8.1 Open and closed sets in Euclidean space

8.2 Closed matrix groups

8.3 Continuous functions

8.4 Compact sets

8.5 Continuous functions and compactness

8.6 Paths and path-connectedness

8.7 Simple connectedness

8.8 Discussion

9 Simply connected Lie groups

9.1 Three groups with tangent space R

9.2 Three groups with the cross-product Lie algebra

9.3 Lie homomorphisms

9.4 Uniform continuity of paths and deformations

9.5 Deforming a path in a sequence of small steps

9.6 Lifting a Lie algebra homomorphism

9.7 Discussion

Bibliography

Index

文摘

Geometry of complex

numbers and quaternions

PREVIEW

When the plane is viewed as the plane C： of complex numbers， rotation about O through angle θ is the same as multiplication by the numbere iθ=cosθ+isinθ．

The set of all such numbers is the unit circle or 1－dimensional sphere

S1={z：|z|=1}．

Thus S1 is not only a geometric object， but also an algebraic structure；in this case a group， under the operation of complex number multiplication．Moreover， the multiplication operation eie1 ．eie2=ei（θ1+θ2）， and the inverse operation （eiθ）－1=ei（－θ）， depend smoothly on the parameter θ．This makes S1 an example of what we call a Lie group．

However， in some respects S1 is too special to be a good illustration of Lie theory． The group S1 is 1－dimensional and commutative，because multiplication of complex numbers is commutative． This property of complex numbers makes the Lie theory of S1 trivial in many ways．

To obtain a more interesting Lie group， we define the four－dimensional algebra of quaternions and the three－dimensional sphere S3 of unit quaternions． Under quaternion multi licaiion， S3 is a noncommutative Lie group known as SU（2）， closely related to the group of space rotations．

1．1 Rotations of the plane

A rotation of the plane R2 about the origin O through angle θ is a linear transformation Re that sends the basis vectors （1，0） and （0，1） to（cosθ，sinθ） and （－sinθ，cosθ）， respectively （Figure 1．1）．

Figure 1．1： Rotation of the plane through angle θ．

It follows by linearity that Re sends the general vector

（x， y）=x（1，0）+y（0， 1） to （xcosθ－ysinθ，xsinθ+ycosθ），and that Re is represented by the matrix

We also call this matrix Rθ． Then applying the rotation to （x， y） is the same as multiplying the column vector （xy） on the left by matrix Rθ， because

This algebraic argument has surprising geometric consequences； forexample， a filling of S3 by disjoint circles known as the Hopf fibration．Figure 2．2 shows some of the circles， projected stereographically into R3．The circles fill nested torus surfaces， one of which is shown in gray． Figure 2．2： Some circles in the Hopf fibration．

Proposition： S3 can be decomposed into disjoint congruent circles．

Proof． As we saw in Section 1．3， the quaternions a+bi+cj+dk of unit length satisfy

a2+b2+C2+d2=1，

and hence they form a 3－sphere S3． The unit quatemions also form a groupG， because the product and inverse of unit quaternions are also unit quaternions， by the multiplicative property of absolute value．

One subgroup H of G consists of the unit quaternions of the formcos O+isin 8， and these form a unit circle in the plane spanned by 1 andi． It follows that any coset qH is also a unit circle， because multiplication by a quaternion q of unit length is an isometry， as we saw in Section 1．4． Since the cosets qH fill the whole group and are disjoint， we have adecomposition of the 3－sphere into unit circles．