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Hardcover ISBN:  9781470474997 
Product Code:  GSM/248 
List Price:  $135.00 
MAA Member Price:  $121.50 
AMS Member Price:  $108.00 
Softcover ISBN:  9781470479152 
Product Code:  GSM/248.S 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470479169 
Product Code:  GSM/248.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470479152 
eBook ISBN:  9781470479169 
Product Code:  GSM/248.S.B 
List Price:  $174.00 $131.50 
MAA Member Price:  $156.60 $118.35 
AMS Member Price:  $139.20 $105.20 

Book DetailsGraduate Studies in MathematicsVolume: 248; 2024; Estimated: 514 ppMSC: Primary 17
Being both a beautiful theory and a valuable tool, Lie algebras form a very important area of mathematics. This modern introduction targets entrylevel graduate students. It might also be of interest to those wanting to refresh their knowledge of the area and be introduced to newer material. Infinitedimensional algebras are treated extensively along with the finitedimensional ones.
After some motivation, the text gives a detailed and concise treatment of the Killing–Cartan classification of finitedimensional semisimple algebras over algebraically closed fields of characteristic 0. Important constructions such as Chevalley bases follow. The second half of the book serves as a broad introduction to algebras of arbitrary dimension, including Kac–Moody (KM), loop, and affine KM algebras. Finitedimensional semisimple algebras are viewed as KM algebras of finite dimension, their representation and character theory developed in terms of integrable representations. The text also covers triangular decomposition (after Moody and Pianzola) and the BGG category \(\mathcal{O}\). A lengthy chapter discusses the Virasoro algebra and its representations. Several applications to physics are touched on via differential equations, Lie groups, superalgebras, and vertex operator algebras.
Each chapter concludes with a problem section and a section on context and history. There is an extensive bibliography, and appendices present some algebraic results used in the book.
ReadershipUndergraduate and graduate students and researchers interested in learning and teaching representations of finitedimensional and infinitedimensional Lie algebras.

Table of Contents

Part I. Preliminaries

Algebras

Examples of Lie algebras

Lie groups

Part II. Classification

Lie algebra basics

The Cartan decomposition

Semisimple Lie algebras: Basic structure

Classification of root systems

Semisimple Lie algebras: Classification

Part III. Important constructions

Finite degree representations of $\mathfrak{sl}_2(\mathbb{K})$

PBW and free Lie algebras

Casimir operators and Weyl’s Theorem II

Chevalley bases and integration

Kac–Moody Lie algebras

Part IV. Representation

Integrable representations

The spherical case and Serre’s Theorem

Irreducible weight modules for $\mathfrak{sl}_2(\mathbb{K})$

Part V. Infinite dimension

Some infinitedimensional Lie algebras

Triangular decomposition and category $\mathcal{O}$

Character theory

Representation of the Virasoro algebra

Part VI. Appendices

Appendix A. Algebra basics

Appendix B. Bilinear forms

Appendix C. Finite groups generated by reflections

Bibliography

Index


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Being both a beautiful theory and a valuable tool, Lie algebras form a very important area of mathematics. This modern introduction targets entrylevel graduate students. It might also be of interest to those wanting to refresh their knowledge of the area and be introduced to newer material. Infinitedimensional algebras are treated extensively along with the finitedimensional ones.
After some motivation, the text gives a detailed and concise treatment of the Killing–Cartan classification of finitedimensional semisimple algebras over algebraically closed fields of characteristic 0. Important constructions such as Chevalley bases follow. The second half of the book serves as a broad introduction to algebras of arbitrary dimension, including Kac–Moody (KM), loop, and affine KM algebras. Finitedimensional semisimple algebras are viewed as KM algebras of finite dimension, their representation and character theory developed in terms of integrable representations. The text also covers triangular decomposition (after Moody and Pianzola) and the BGG category \(\mathcal{O}\). A lengthy chapter discusses the Virasoro algebra and its representations. Several applications to physics are touched on via differential equations, Lie groups, superalgebras, and vertex operator algebras.
Each chapter concludes with a problem section and a section on context and history. There is an extensive bibliography, and appendices present some algebraic results used in the book.
Undergraduate and graduate students and researchers interested in learning and teaching representations of finitedimensional and infinitedimensional Lie algebras.

Part I. Preliminaries

Algebras

Examples of Lie algebras

Lie groups

Part II. Classification

Lie algebra basics

The Cartan decomposition

Semisimple Lie algebras: Basic structure

Classification of root systems

Semisimple Lie algebras: Classification

Part III. Important constructions

Finite degree representations of $\mathfrak{sl}_2(\mathbb{K})$

PBW and free Lie algebras

Casimir operators and Weyl’s Theorem II

Chevalley bases and integration

Kac–Moody Lie algebras

Part IV. Representation

Integrable representations

The spherical case and Serre’s Theorem

Irreducible weight modules for $\mathfrak{sl}_2(\mathbb{K})$

Part V. Infinite dimension

Some infinitedimensional Lie algebras

Triangular decomposition and category $\mathcal{O}$

Character theory

Representation of the Virasoro algebra

Part VI. Appendices

Appendix A. Algebra basics

Appendix B. Bilinear forms

Appendix C. Finite groups generated by reflections

Bibliography

Index