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Author:

Erdmann, Karin, 1948-

Title:

Introduction to Lie algebras / Karin Erdmann and Mark J. Wildon.

Publisher:

Springer,

Copyright Date:

℗♭2006

Description:

x, 251 pages : illustrations ; 24 cm.

Subject:

Lie algebras.

Other Authors:

Wildon, Mark J.

Notes:

Includes bibliographical references (pages 247-248) and index.

Contents:

231. Appendix E: Answers to Selected Exercises 1 -- 1.3 Subalgebras and Ideals 3 -- 1.4 Homomorphisms 4 -- 1.5 Algebras 5 -- 1.6 Derivations 6 -- 1.7 Structure Constants 7 -- 2 Ideals and Homomorphisms 11 -- 2.1 Constructions with Ideals 11 -- 2.2 Quotient Algebras 12 -- 2.3 Correspondence between Ideals 14 -- 3 Low-Dimensional Lie Algebras 19 -- 3.1 Dimensions 1 and 2 20 -- 3.2 Dimension 3 20 -- 4 Solvable Lie Algebras and a Rough Classification 27 -- 4.1 Solvable Lie Algebras 27 -- 4.2 Nilpotent Lie Algebras 31 -- 4.3 A Look Ahead 32 -- 5 Subalgebras of gl(V) 37 -- 5.1 Nilpotent Maps 37 -- 5.2 Weights 38 -- 5.3 The Invariance Lemma 39 -- 5.4 An Application of the Invariance Lemma 42 -- 6 Engel's Theorem and Lie's Theorem 45 -- 6.1 Engel's Theorem 46 -- 6.2 Proof of Engel's Theorem 48 -- 6.3 Another Point of View 48 -- 6.4 Lie's Theorem 49 -- 7 Some Representation Theory 53 -- 7.2 Examples of Representations 54 -- 7.3 Modules for Lie Algebras 55 -- 7.4 Submodules and Factor Modules 57 -- 7.5 Irreducible and Indecomposable Modules 58 -- 7.6 Homomorphisms 60 -- 7.7 Schur's Lemma 61 -- 8 Representations of sl(2, C) 67 -- 8.1 The Modules V[subscript d] 67 -- 8.2 Classifying the Irreducible sl(2, C)-Modules 71 -- 8.3 Weyl's Theorem 74 -- 9 Cartan's Criteria 77 -- 9.1 Jordan Decomposition 77 -- 9.2 Testing for Solvability 78 -- 9.3 The Killing Form 80 -- 9.4 Testing for Semisimplicity 81 -- 9.5 Derivations of Semisimple Lie Algebras 84 -- 9.6 Abstract Jordan Decomposition 85 -- 10 The Root Space Decomposition 91 -- 10.1 Preliminary Results 92 -- 10.2 Cartan Subalgebras 95 -- 10.3 Definition of the Root Space Decomposition 97 -- 10.4 Subalgebras Isomorphic to sl(2, C) 97 -- 10.5 Root Strings and Eigenvalues 99 -- 10.6 Cartan Subalgebras as Inner-Product Spaces 102 -- 11 Root Systems 109 -- 11.1 Definition of Root Systems 109 -- 11.2 First Steps in the Classification 111 -- 11.3 Bases for Root Systems 115 -- 11.4 Cartan Matrices and Dynkin Diagrams 120 -- 12 The Classical Lie Algebras 125 -- 12.1 General Strategy 126 -- 12.2 sl(l + 1, C) 129 -- 12.3 so(2l + 1, C) 130 -- 12.4 so(2l, C) 133 -- 12.5 sp(2l, C) 134 -- 12.6 Killing Forms of the Classical Lie Algebras 136 -- 12.7 Root Systems and Isomorphisms 137 -- 13 The Classification of Root Systems 141 -- 13.1 Classification of Dynkin Diagrams 142 -- 13.2 Constructions 148 -- 14 Simple Lie Algebras 153 -- 14.1 Serre's Theorem 154 -- 14.2 On the Proof of Serre's Theorem 158 -- 15 Further Directions 163 -- 15.1 The Irreducible Representations of a Semisimple Lie Algebra 164 -- 15.2 Universal Enveloping Algebras 171 -- 15.3 Groups of Lie Type 177 -- 15.4 Kac-Moody Lie Algebras 179 -- 15.5 The Restricted Burnside Problem 180 -- 15.6 Lie Algebras over Fields of Prime Characteristic 183 -- 15.7 Quivers 184 -- 16 Appendix A: Linear Algebra 189 -- 16.1 Quotient Spaces 189 -- 16.2 Linear Maps 191 -- 16.3 Matrices and Diagonalisation 192 -- 16.4 Interlude: The Diagonal Fallacy 197 -- 16.5 Jordan Canonical Form 198 -- 16.6 Jordan Decomposition 200 -- 16.7 Bilinear Algebra 201 -- 17 Appendix B: Weyl's Theorem 209 -- 17.1 Trace Forms 209 -- 17.2 The Casimir Operator 211 -- 18 Appendix C: Cartan Subalgebras 215 -- 18.1 Root Systems of Classical Lie Algebras 215 -- 18.2 Orthogonal and Symplectic Lie Algebras 217 -- 18.3 Exceptional Lie Algebras 220 -- 18.4 Maximal Toral Subalgebras 221 -- 19 Appendix D: Weyl Groups 223 -- 19.1 Proof of Existence 223 -- 19.2 Proof of Uniqueness 224 -- 19.3 Weyl Groups 226 -- 20 Appendix E: Answers to Selected Exercises 231.

Series:

Springer undergraduate mathematics series

ISBN:

1846280400
9781846280405

OCLC:

(OCoLC)63137193

LCCN:

2005937687

Locations:

USUX851 -- Iowa State University - Parks Library (Ames)

OVUX522 -- University of Iowa Libraries (Iowa City)