Lie groups and algebras with applications to physics, geometry, and mechanics
The theory of Lie groups and algebras has a beautiful and subtle mathematical structure and important applications. Particle physics and gauge field theory, symmetries in quantum mechanics and chemistry, completely integrable systems and geometric control theory are the fertile areas for such applications. There are excellent books like \textit{S. Helgason}'s ''Differential geometry, Lie groups and symmetric spaces'' (1978; Zbl 0451.53038) which treat the structure theory of Lie groups and algebras and their mathematical applications. But certainly the books of this kind are too difficult for an application oriented reader. On the other hand, ''Group theory in physics''-type books deal with one sort of computations or another rather than the structural aspects of the theory. The book under review is one of the first attempts to fill in the expository gap in this area. It provides a rigorous introduction to the most fundamental aspects of the structure theory, including the classification of semisimple Lie algebras over \({\mathbb{C}}\), Cartan criterion for the semisimplicity, Cartan decomposition, the correspondence between Lie groups and Lie algebras. A necessary background in differential manifolds and differential geometry is given. On the other hand, an impressive number of concrete examples and applications are also considered. The introductory chapter deals with the description of SU(2) as a covering group of SO(3), the Lorenz group, and contains a brief discussion of the classical groups. We must mention symmetry groups of differential equations, completely integrable systems (Toda lattice and Korteweg-de Vries hierarchy), a spontaneous symmetry breaking, Boson realizations of Lie algebras as the examples of applications considered in this book. The book is very well written and can be recommended to an application oriented reader.
symmetry groups of differential equations
scientific article; zbMATH DE number 3957480