Representation Theory of Locally Compact Abelian Groups and Compact Groups

Raychaudhuri, Arani (2022) Representation Theory of Locally Compact Abelian Groups and Compact Groups. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS dissertation of Arani Raychaudhuri (16MS148)) Thesis_16MS148.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

The goal of this project is to get an introductory understanding of the vast field of Abstract Harmonic Analysis, in this case, with emphasis on Representation Theory. We present some important results from Functional Analysis and Operator Theory in the first chapter, moving on to the understanding of Haar Meaure and the analogous Modular Function and Convolutions of Functions and Measures in the second chapter. We study the naturally consequent topics of Unitary Representations and prove Gelfand-Raikov Theorem in the third chapter and the resulting structures and existance of Unitary Representations. The final two chapters are my foray into the study of Representations (Unitary) of Topological Groups. First of them consists of the Representation of Locally Compact Abelian Groups, which includes the construction of Fourier Transform, Fourier Inversion and Pontrjagin Duality, which is utterly important to identify a Locally Compact Group with the Dual of its Dual Group and the chapter ends with insights of Representation Theory into it. And, the second one of them develops the theory of Unitary Representations of Compact Groups, the decomposition of L2(G) by Peter-Weyl Theorem and the construction of Fourier Transform on L2(G) and finally, I concluded it with understanding some examples of Representations on Compact Groups (Non-Abelian) such as SU(2), SO(3) and SO(4). vii

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