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Introduction to Lie Algebras: Finite and Infinite Dimension
 
J. I. Hall Michigan State University, East Lansing, MI
Hardcover ISBN:  978-1-4704-7499-7
Product Code:  GSM/248
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
Softcover ISBN:  978-1-4704-7915-2
Product Code:  GSM/248.S
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-7916-9
Product Code:  GSM/248.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-7915-2
eBook: ISBN:  978-1-4704-7916-9
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
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Introduction to Lie Algebras: Finite and Infinite Dimension
J. I. Hall Michigan State University, East Lansing, MI
Hardcover ISBN:  978-1-4704-7499-7
Product Code:  GSM/248
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
Softcover ISBN:  978-1-4704-7915-2
Product Code:  GSM/248.S
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-7916-9
Product Code:  GSM/248.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-7915-2
eBook ISBN:  978-1-4704-7916-9
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 Details
     
     
    Graduate Studies in Mathematics
    Volume: 2482024; 514 pp
    MSC: Primary 17

    Being both a beautiful theory and a valuable tool, Lie algebras form a very important area of mathematics. This modern introduction targets entry-level graduate students. It might also be of interest to those wanting to refresh their knowledge of the area and be introduced to newer material. Infinite dimensional algebras are treated extensively along with the finite dimensional ones.

    After some motivation, the text gives a detailed and concise treatment of the Killing–Cartan classification of finite dimensional 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. Finite dimensional 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.

    Readership

    Undergraduate and graduate students and researchers interested in learning and teaching representations of finite-dimensional and infinite-dimensional 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 infinite dimensional 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 2482024; 514 pp
MSC: Primary 17

Being both a beautiful theory and a valuable tool, Lie algebras form a very important area of mathematics. This modern introduction targets entry-level graduate students. It might also be of interest to those wanting to refresh their knowledge of the area and be introduced to newer material. Infinite dimensional algebras are treated extensively along with the finite dimensional ones.

After some motivation, the text gives a detailed and concise treatment of the Killing–Cartan classification of finite dimensional 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. Finite dimensional 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.

Readership

Undergraduate and graduate students and researchers interested in learning and teaching representations of finite-dimensional and infinite-dimensional 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 infinite dimensional 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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
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