Groups and Manifolds: Lectures for Physicists with Examples in Mathematica

Description

Groups and Manifolds is an introductory, yet a complete self-contained course on mathematics of symmetry: group theory and differential geometry of symmetric spaces, with a variety of examples for physicists, touching briefly also on super-symmetric field theories. The core of the course is focused on the construction of simple Lie algebras, emphasizing the double interpretation of the ADE classification as applied to finite rotation groups and to simply laced simple Lie algebras. Unique features of this book are the full-fledged treatment of the exceptional Lie algebras and a rich collection of MATHEMATICA Notebooks implementing various group theoretical constructions. From basic Algebra to Super-symmetry. Uses Mathematica (code available online) for group-theoretical constructions. Includes a variety of physical examples from crystallography, atomic physics, cosmology and field theory. Preface Acknowledgement Contents Some remarks about notation 1. Fundamental notions of algebra 2. Groups: a bestiary for beginners 3. Basic elements of finite group theory 4. Finite subgroups of SO(3) and crystallographic groups 5. Manifolds and Lie groups 6. Structure of Lie algebras 7. Root systems and their classification 8. Lie algebra representation theory 9. Exceptional Lie algebras 10. A primary on the theory of connections and metrics 11. Isometries and the geometry of coset manifolds 12. An anthology of group theory physical applications A. Exercises B. MATHEMATICA NoteBooks Bibliography