Groups and symmetries : from finite groups to Lie groups

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Groups and symmetries : from finite groups to Lie groups / Yvette Kosmann-Schwarzbach ; translated by Stephanie Frank Singer

資料種別:

図書

出版情報:

New York : Springer, c2010

形態:

xv, 194 p. ; 24 cm

シリーズ名:

Universitext <BA00021004>

著者名:

ISBN:

9780387788654 [0387788654]

書誌ID:

BA91908509

子書誌情報

フルテキスト

所蔵情報

他の版・巻

書誌詳細

注記:

Includes bibliographical references (p. [185]-189) and index

主題:

現代代数学．抽象代数学．群論

言語:

英語

目次情報:

Introduction

Acknowledgments

General Facts About Groups / 1：

Review of Definitions

Examples of Finitep2 / 2：

Cyclic Group of Order n / 2.1：

Symmetric Group ©n / 2.2：

Dihedral Group / 2.3：

Other Examples / 2.4：

Examples of Infinite Groups / 3：

Group Actions and Conjugacy Classes / 4：

References

Exercises

Representations of Finite Groups

Representations

General Facts / 1.1：

Irreducible Representations / 1.2：

Direct Sum of Representations / 1.3：

Intertwining Operators and Schur's Lemma / 1.4：

Characters and Orthogonality Relations

Functions on a Group, Matrix Coefficients

Characters of Representations and Orthogonality Relations

Character Table

Application to the Decomposition of Representations

The Regular Representation

Definition / 3.1：

Character of the Regular Representation / 3.2：

Isotypic Decomposition / 3.3：

Basis of the Vector Space of Class Functions / 3.4：

Projection Operators

Induced Representations / 5：

Geometric Interpretation / 5.1：

Representations of Compact Groups

Compact Groups

Haar Measure

Representations of Topological Groups and Schur's Lemma

Coefficients of a Representation

Intertwining Operators

Operations on Representations

Schur's Lemma / 3.5：

Complete Reducibility / 4.1：

Orthogonality Relations / 4.2：

Summary of Chapter 3

Lie Groups and Lie Algebras

Lie Algebras

Definition and Examples

Morphisms

Commutation Relations and Structure Constants

Real Forms

Representations of Lie Algebras / 1.5：

Review of the Exponential Map

One-Parameter Subgroups of GL(n, K)

Lie Groups

The Lie Algebra of a Lie Group

The Connected Component of the Identity / 6：

Morphisms of Lie Groups and of Lie Algebras / 7：

Differential of a Lie Group Morphism / 7.1：

Differential of a Lie Group Representation / 7.2：

The Adjoint Representation / 7.3：

Lie Groups SU(2) and SO(3)

The Lie Algebras su(2) and so(3)

Bases of su(2)

Bases of so(3)

Bases of s1(2, C)

The Covering Morphism of SU(2) onto SO(3)

The Lie Group SO(3)

The Lie Group SU(2)

Projection of SU(2) onto SO(3)

Representations of SU(2) and SO(3)

Irreducible Representations of s1(2,C)

The Representations Dj

The Casimir Operator

Hermitian Nature of the Operators J3 and J2

Representations of SU(2)

The Representations TP

Characters of the Representations Dj

Representations of SO(3)

Spherical Harmonics

Review of L2(S2)

Harmonic Polynomials

Representations of Groups on Function Spaces

Spaces of Harmonic Polynomials

Representations of SO(3) on Spaces of Harmonic Polynomials

Definition of Spherical Harmonics

Representations of SO(3) on Spaces of Spherical Harmonics

Eigenfunctions of the Casimir Operator

Bases of Spaces of Spherical Harmonics

Explicit Formulas

Representations of SU(3) and Quarks / 8：

Review of s1(n,C), Representations of s1(3,C) and SU(3)

Review of s l (n, C)

The Case of s1(3, C)

The Bases (I3,Y) and (I3,TB) of h

Representations of sl (3,C) and of SU(3)

The Adjoint Representation and Roots

The Fundamental Representation and Its Dual

The Fundamental Representation

The Dual of the Fundamental Representation

Highest Weight of a Finite-Dimensional Representation

Highest Weight

Weights as Linear Combinations of the &lamda;_i

Finite-Dimensional Representations and Weights / 4.3：

Another Example: The Representation 6 / 4.4：

One More Example: The Representation 10 / 4.5：

Tensor Products of Representations

The Eightfold Way

Baryons (B=1) / 6.1：

Mesons (B=0) / 6.2：

Baryon Resonances / 6.3：

Problems and Solutions

Restriction of a Representation to a Finite Groups

The Group O(2)

Representation of the Dihedral and Quaternion Groups

Irreducible Representations of SU(2) and of G₃

Pseudo-unitary and Pseudo-orthogonal Groups

Irreducible Representations of SU(2)x SU(2)

Symmetries of Fullerene Molecules

Matrix Coefficients and Spherical Harmonics / 9：

Bibliography

Index

Introduction

Acknowledgments

General Facts About Groups / 1：

Review of Definitions

Examples of Finitep2 / 2：

Cyclic Group of Order n / 2.1：

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