Symmetries, Lie algebras and representations. A graduate course for physicists.

*(English)*Zbl 0923.17001
Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. xxi, 438 p. (1997).

This book is an introduction to the theory of Lie algebras, their representations, and their applications in physics.

The first three chapters show how Lie algebras arise naturally from symmetries of physical systems. Through the examples of the algebra of angular momentum in quantum theory, the free scalar field, the Heisenberg algebra, and the Lie algebra of \(su(3)\) and hadron symmetries, much of the general structure of the theory of Lie algebras and their representations is first illustrated by means of examples familiar to physicists.

Chapters 4 to 13 give a detailed introduction to Lie algebras and their representations. First, Lie algebras are defined in an abstract way, but soon the relation with physics is made by having Lie algebras act in some vector space, leading to the notion of representation. Then some structure theory of Lie algebras, covering the Cartan-Weyl basis, is developed. For simple Lie algebras, Dynkin diagrams and the root structure are described, and an introduction to affine Lie algebras is given. Topics such as real Lie algebras and real forms, Lie groups, the Weyl group (finite and affine case), automorphisms, loop algebras and central extensions are treated in separate chapters. The chapter on highest weight representations deals with weight systems, the Weyl dimension and character formula.

Chapters 14 to 22 cover specific further topics, such as Verma modules, Casimirs, tensor products and Clebsch-Gordan coefficients, invariant tensors, subalgebras and branching rules, Young tableaux, spinors, Clifford algebras and supersymmetry, representations (of Lie groups) on function spaces, and Hopf algebras and representation rings.

A detailed reference list is provided. Every chapter ends with a short and handy summary, a set of keywords, and a list of exercises. Many exercises and examples throughout the book illustrate the use of Lie algebras in real physical problems.

The text is written at a level accessible to graduate students, and provides a comprehensive reference list useful for researchers.

The first three chapters show how Lie algebras arise naturally from symmetries of physical systems. Through the examples of the algebra of angular momentum in quantum theory, the free scalar field, the Heisenberg algebra, and the Lie algebra of \(su(3)\) and hadron symmetries, much of the general structure of the theory of Lie algebras and their representations is first illustrated by means of examples familiar to physicists.

Chapters 4 to 13 give a detailed introduction to Lie algebras and their representations. First, Lie algebras are defined in an abstract way, but soon the relation with physics is made by having Lie algebras act in some vector space, leading to the notion of representation. Then some structure theory of Lie algebras, covering the Cartan-Weyl basis, is developed. For simple Lie algebras, Dynkin diagrams and the root structure are described, and an introduction to affine Lie algebras is given. Topics such as real Lie algebras and real forms, Lie groups, the Weyl group (finite and affine case), automorphisms, loop algebras and central extensions are treated in separate chapters. The chapter on highest weight representations deals with weight systems, the Weyl dimension and character formula.

Chapters 14 to 22 cover specific further topics, such as Verma modules, Casimirs, tensor products and Clebsch-Gordan coefficients, invariant tensors, subalgebras and branching rules, Young tableaux, spinors, Clifford algebras and supersymmetry, representations (of Lie groups) on function spaces, and Hopf algebras and representation rings.

A detailed reference list is provided. Every chapter ends with a short and handy summary, a set of keywords, and a list of exercises. Many exercises and examples throughout the book illustrate the use of Lie algebras in real physical problems.

The text is written at a level accessible to graduate students, and provides a comprehensive reference list useful for researchers.

Reviewer: J.Van der Jeugt (Gent)

##### MSC:

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |

81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |

17B81 | Applications of Lie (super)algebras to physics, etc. |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

22E60 | Lie algebras of Lie groups |

17B05 | Structure theory for Lie algebras and superalgebras |

17B20 | Simple, semisimple, reductive (super)algebras |