Naive Lie Theory _生活百科

Naive Lie Theory【Naive Lie Theory】2009年Springer New York出版社出版John Stillwell编着图书。
基本介绍中文名称：Naive Lie Theory
装帧：Paperback
定价：USD 49.95
作者：John Stillwell
出版社：Springer New York
出版日期：2009-12-28
ISBN：9781441926814
编辑推荐《朴素李理论》是一部介绍李群和李代数的本科生教程，基本的微积分和线性代数知识将对理解《朴素李理论》十分重要。为了让更多读者受益，也是《朴素李理论》的最直接目的，书中对经典群核实、复和四元数空间做了较深刻地介绍。书中从矩阵的角度讲述对称群，这样就可以用微积分和线性代数的基础理论理解《朴素李理论》中的内容。《朴素李理论》由（美）史迪威着。目录1Geometry of complex numbers and quaternions1.1Rotations of the plane1.2Matrix representation of complex numbers1.3Quaternions1.4Consequences of multiplicative absolute value1.5Quaternion representation of space rotations1.6Discussion2Groups2.1Crash course on groups2.2Crash course on homomorphisms2.3The groups SU(2) and SO(3)2.4Isometrics of R'' and reflections2.5Rotations of R4 and pairs of quaternions2.6Direct products of groups2.7The map from SU(2)SU(2) to SO(4)2.8Discussion3Generalized rotation groups3.1Rotations as orthogonal transformations3.2The orthogonai and special orthogonal groups3.3The unitary groups3.4The symplectic groups3.5Maximal tori and centers3.6Maximal tori in SO(n), U(n), SU(n), Sp(n)3.7Centers of SO(n), U(n), SU(n), Sp(n)3.8Connectcdness and discreteness3.9Discussion4The exponential map4.1The exponential map onto SO(2)4.2The exponential map onto SU(2)4.3The tangent space of SU(2)4.4The Lie algebra su(2) of SU(2)4.5The exponential of a square matrix4.6The affine group of the line4.7Discussion5The tangent space5.1Tangent vectors of O(n), U(n), Sp(n)5.2The tangent space of SO(n)5.3The tangent space of U(n), SU(n), Sp(n)5.4Algebraic properties of the tangent space5.5Dimension of Lie algebras5.6Complexification5.7Quaternion Lie algebras5.8Discussion6Structure of Lie algebras6.1Normal subgroups and ideals6.2Ideals and homomorphisms6.3Classical non-simple Lie algebras6.4Simplicity of (n,C) and su(n)6.5Simplicity of o(n) for n > 46.6Simplicity of p(n)6.7Discussion7The matrix logarithm7.1Logarithm and exponential7.2The exp function on the tangent space7.3Limit properties of log and exp7.4The log function into the tangentspace7.5SO(n), SU(n), and Sp(n) revisited7.6The Campbell-Baker-Hausdorff theorem7.7Eichler's proof of Campbell-Baker-Hausdorff7,8Discussion8 Topology8.1Open and closed sets in Euclidean space8.2Closed matrix groups8.3Continuous functions8.4Compact sets8.5Continuous functions and compactness8.6Paths and path-connectedness 8.7Simple connectedness8.8Discussion9Simply connected Lie groups9.1Three groups with tangent space R9.2Three groups with the cross-product Lie algebra9.3Lie homomorphisms9.4Uniform continuity of paths and deformations9.5Deforming a path in a sequence of small steps9.6Lifting a Lie algebra homomorphism9.7DiscussionBibliographyIndex文摘Geometry of complexnumbers and quaternionsPREVIEWWhen 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 sphereS1={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 planeA 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 matrixWe 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，becauseThis 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 satisfya2+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．