This book, "Introduction to Vertex Operator Algebras and Their Representations," is authored by James Lepowsky and Haisheng Li and published by Springer Science+Business Media, LLC. It is part of the "Progress in Mathematics" series, volume 227, and serves as a comprehensive introduction to the theory of vertex operator algebras (VOAs) and their representations. The book begins with an introduction that motivates the study of VOAs and provides an overview of the content. It covers formal calculus, the axiomatic foundations of VOAs, and the construction of examples. Key topics include: 1. **Formal Calculus**: Introduces formal series, the formal delta function, derivations, and the formal Taylor theorem. 2. **Axiomatic Basics**: Discusses definitions, commutativity, associativity, and the Jacobi identity. 3. **Modules**: Explores the definition and properties of modules for VOAs. 4. **Representations and Construction**: Focuses on weak vertex operators, the canonical weak vertex algebra, and the construction of vertex algebras and modules. 5. **Construction of Families**: Provides detailed constructions of various families of VOAs and modules, including those associated with the Virasoro algebra, affine Lie algebras, Heisenberg algebras, and even lattices. The authors emphasize the axiomatic approach, which is a significant departure from earlier treatments that focused on examples. They also highlight the importance of vertex operator algebras in understanding "monstrous moonshine" and their applications in string theory and infinite-dimensional Lie algebra theory. The book is suitable for graduate students and researchers in mathematics and theoretical physics, providing a self-contained introduction to the theory of vertex operator algebras and their applications. It includes new results and advanced treatments, making it a valuable resource for both beginners and experts in the field.This book, "Introduction to Vertex Operator Algebras and Their Representations," is authored by James Lepowsky and Haisheng Li and published by Springer Science+Business Media, LLC. It is part of the "Progress in Mathematics" series, volume 227, and serves as a comprehensive introduction to the theory of vertex operator algebras (VOAs) and their representations. The book begins with an introduction that motivates the study of VOAs and provides an overview of the content. It covers formal calculus, the axiomatic foundations of VOAs, and the construction of examples. Key topics include: 1. **Formal Calculus**: Introduces formal series, the formal delta function, derivations, and the formal Taylor theorem. 2. **Axiomatic Basics**: Discusses definitions, commutativity, associativity, and the Jacobi identity. 3. **Modules**: Explores the definition and properties of modules for VOAs. 4. **Representations and Construction**: Focuses on weak vertex operators, the canonical weak vertex algebra, and the construction of vertex algebras and modules. 5. **Construction of Families**: Provides detailed constructions of various families of VOAs and modules, including those associated with the Virasoro algebra, affine Lie algebras, Heisenberg algebras, and even lattices. The authors emphasize the axiomatic approach, which is a significant departure from earlier treatments that focused on examples. They also highlight the importance of vertex operator algebras in understanding "monstrous moonshine" and their applications in string theory and infinite-dimensional Lie algebra theory. The book is suitable for graduate students and researchers in mathematics and theoretical physics, providing a self-contained introduction to the theory of vertex operator algebras and their applications. It includes new results and advanced treatments, making it a valuable resource for both beginners and experts in the field.