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Covering a wide range of material, this volume describes fundamental aspects of representation theory of the Virasoro algebra. Topics include the structure of Verma modules and Fock modules, the classification of (unitarizable) Harish-Chandra modules, tilting equivalence, and more.

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The Virasoro algebra is an infinite dimensional Lie algebra that plays an increasingly important role in mathematics and theoretical physics. This book describes some fundamental facts about the representation theory of the Virasoro algebra in a self-contained manner. Topics include the structure of Verma modules and Fock modules, the classification of (unitarizable) Harish-Chandra modules, tilting equivalence, and the rational vertex operator algebras associated to the so-called minimal series representations.

Covering a wide range of material, this book has three appendices which provide background information required for some of the chapters. The authors organize fundamental results in a unified way and refine existing proofs. For instance in chapter three, a generalization of Jantzen filtration is reformulated in an algebraic manner, and geometric interpretation is provided. Statements, widely believed to be true, are collated, and results which are known but not verified are proven, such as the corrected structure theorem of Fock modules in chapter eight.

This book will be of interest to a wide range of mathematicians and physicists from the level of graduate students to researchers.

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Preliminary.- Classification of Harish-Chandra Modules.- The Jantzen Filtration.- Determinant Formulae.- Verma Modules I: Preliminaries.- Verma Modules II: Structure Theorem.- A Duality among Verma Modules.- Fock Modules.- Rational Vertex Operator Algebras.- Coset Constructions for sl2.- Unitarisable Harish-Chandra Modules.- Homological Algebras.- Lie p-algebras.- Vertex Operator Algebras.

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From the reviews:

"The book under review collects many classical results on representations of the Virasoro algebra and can be used both as a reference source and as a textbook." (Volodymyr Mazorchuk, Mathematical Reviews, Issue 2011 m)

"The aim of the book is to describe fundamental facts about the representation theory of the Virasoro algebras in a self-contained manner. ... At the end of each chapter are bibliographical notes and comments, and some times some extra appendices. A quick guide for further reading of some topics that are not treated in this book is presented, as well as a comprehensive list of references." (Daniel Bulacu, Zentralblatt MATH, Vol. 1222, 2011)

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