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Representations of Lie algebras : an introduction through gln

Author: Anthony Henderson
Publisher: Cambridge : Cambridge University Press, 2012.
Series: Australian Mathematical Society lecture series, 22.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
"This bold and refreshing approach to Lie algebras assumes only modest prerequisites (linear algebra up to the Jordan canonical form and a basic familiarity with groups and rings), yet it reaches a major result in representation theory: the highest-weight classification of irreducible modules of the general linear Lie algebra. The author's exposition is focused on this goal rather than aiming at the widest  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Henderson, Anthony, 1976-
Representations of Lie algebras
(DLC) 2012021841
(OCoLC)785872028
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Anthony Henderson
ISBN: 9781139550208 1139550209 1139555162 9781139555166 9781139236126 1139236121 9781139564984 1139564986
OCLC Number: 808501344
Description: 1 online resource (ix, 156 pages) : illustrations.
Contents: Cover; Representations of Lie Algebras; AUSTRALIAN MATHEMATICAL SOCIETY LECTURE SERIES; Title; Copyright; Contents; Preface; Notational conventions; CHAPTER 1 Motivation: representations of Lie groups; 1.1 Homomorphisms of general linear groups; 1.2 Multilinear algebra; 1.3 Linearization of the problem; 1.4 Lie's theorem; CHAPTER 2 Definition of a Lie algebra; 2.1 Definition and first examples; 2.2 Classification and isomorphisms; 2.3 Exercises; CHAPTER 3 Basic structure of a Lie algebra; 3.1 Lie subalgebras; 3.2 Ideals; 3.3 Quotients and simple Lie algebras; 3.4 Exercises. CHAPTER 4 Modules over a Lie algebra; 4.1 Definition of a module; 4.2 Isomorphism of modules; 4.3 Submodules and irreducible modules; 4.4 Complete reducibility; 4.5 Exercises; CHAPTER 5 The theory of sl2-modules; 5.1 Classification of irreducibles; 5.2 Complete reducibility; 5.3 Exercises; CHAPTER 6 General theory of modules; 6.1 Duals and tensor products; 6.2 Hom-spaces and bilinear forms; 6.3 Schur's lemma and the Killing form; 6.4 Casimir operators; 6.5 Exercises; CHAPTER 7 Integral gln-modules; 7.1 Integral weights; 7.2 Highest-weight modules; 7.3 Irreducibility of highest-weight modules. 7.4 Tensor-product construction of irreducibles; 7.5 Complete reducibility; 7.6 Exercises; CHAPTER 8 Guide to further reading; 8.1 Classification of simple Lie algebras; 8.2 Representations of simple Lie algebras; 8.3 Characters and bases of representations; APPENDIX Solutions to the exercises; Solutions for Chapter 2 exercises; Solutions for Chapter 3 exercises; Solutions for Chapter 4 exercises; Solutions for Chapter 5 exercises; Solutions for Chapter 6 exercises; Solutions for Chapter 7 exercises; References; Index.
Series Title: Australian Mathematical Society lecture series, 22.
Responsibility: Anthony Henderson, School of Mathematics and Statistics, University of Sydney.
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Abstract:

A fresh undergraduate-accessible approach to Lie algebras and their representations.  Read more...

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'This short book develops the standard tools (special bases, duality, tensor decompositions, Killing forms, Casimir operators ...) and aims for a single result: the classification of integral modules Read more...

 
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