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This book is intended to serve as a text book for a one-semester course for MSc/MPhil students of mathematics at Indian universities, and has, in fact, been class tested by two of the authors at the Master's level. Most students are not introduced to Lie Theory and non- commutative harmonic analysis until they are in the second year of the PhD programme. In these notes, by sticking to closed subgroups of the general linear group, the authors give a flavour of Lie Theory, while keeping the prerequisites to a minimum. The only prerequisites are Real Analysis (including some Fourier Series) and Elementary Functional Analysis. Students of theoretical physics will also find this exposition useful.
S C Bagchi, Indian Statistical Institute, Calcutta. S Madan, Indian Institute of Technology, Kanpur. A Sitaram, Indian Statistical Institute, Bangalore. U B Tewari, Indian Institute ofTechnology, Kanpur.
Preface / Fourier Series and Fourier Transforms - Fourier Series; Fourier Transforms on R and Rn; The Isoperimetric Problem; Notes / Topological Groups - Definitions and Examples; Haar Measure; Lp Spaces, Convolution and Approximate Identities; Notes / Basic Representation Theory and the Peter-Wey1 Theorem - Basic Representation Theory; Representation Theory of Compact Groups: The Peter-Wey1 Theorem; Irreducible Unitary Representations of SU(2); Notes / Linear Lie Groups – Introduction; The Exponential Map and the Lie Algebra of a Linear Lie Group; Some Odds and Ends; Some Calculus on a Linear Lie Group; Invariant Differential Operators on G; Finite Dimensional Representation of G and g; SU(2) Revisited: Hormonic Analysis on SU(2); Irreducible Unitary Representation of SO(3); Some Examples of Linear Lie Groups and Their Lie Algebras; Notes / The Sphere s2 - Some Harmonic Analysis; Convex Bodies with Sections of Equal Area; Concluding Remarks / References / Further Reading / Index