Introduction to Lie Algebras
American Mathematical Society (Verlag)
978-1-4704-7499-7 (ISBN)
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.
J. I. Hall, Michigan State University, East Lansing, MI
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
Bibliography
Index
| Erscheinungsdatum | 04.03.2025 |
|---|---|
| Reihe/Serie | Graduate Studies in Mathematics |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 1-4704-7499-9 / 1470474999 |
| ISBN-13 | 978-1-4704-7499-7 / 9781470474997 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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